The document is a summary of lecture notes for a Calculus I class. It discusses integration by substitution, providing theory, examples, and objectives. Key points covered include the substitution rule for indefinite integrals, working through examples like finding the integral of √x2+1 dx, and noting substitution can transform integrals into simpler forms. Definite integrals using substitution are also briefly mentioned.
Lesson 27: Integration by Substitution, part II (Section 10 version)Matthew Leingang
The document is the notes for a Calculus I class. It provides announcements for the upcoming class on Monday, which will involve reviewing course material rather than new topics. Examples are given of integration by substitution, including exponential, odd, and even functions. Multiple methods for substitutions are presented. The properties of odd and even functions are defined, and examples are shown graphically. Symmetric functions and their behavior under combinations are discussed.
Lesson 29: Integration by Substition (worksheet solutions)Matthew Leingang
This document contains the notes from a calculus class. It provides announcements about the final exam schedule and review sessions. It then discusses the technique of u-substitution for both indefinite and definite integrals. Examples are provided to illustrate how to use u-substitution to evaluate integrals involving trigonometric, polynomial, and other functions. The document emphasizes that u-substitution often makes evaluating integrals much easier than expanding them out directly.
This document is the outline for a calculus class. It discusses the final exam date and review sessions, and outlines the topics of substitution for indefinite integrals and substitution for definite integrals. It provides an example of using substitution to find the integral of the square root of x^2 + 1 by letting u = x^2 + 1, and expresses this using both standard notation and Leibniz notation. It states the theorem of substitution rule.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 27: Integration by Substitution (Section 041 slides)Matthew Leingang
The document contains notes from a Calculus I class at New York University on December 13, 2010. It discusses using the substitution method for indefinite and definite integrals. Examples are provided to demonstrate how to use substitutions to evaluate integrals involving trigonometric, exponential, and polynomial functions. The key steps are to make a substitution for the variable in terms of a new variable, determine the differential of the substitution, and substitute into the integral to transform it into an integral involving only the new variable.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
Lesson 27: Integration by Substitution (Section 4 version)Matthew Leingang
The document outlines a calculus lecture on integration by substitution. It provides examples of using u-substitution to find antiderivatives of expressions like √(x^2+1) and tan(x). The key ideas are that if u is a function of x, its derivative du/dx can be used to rewrite the integrand and perform a u-substitution integration.
1. The document provides solutions to homework problems involving partial differential equations.
2. Problem 1 solves the wave equation utt = c2uxx using d'Alembert's formula to find the solution u(x,t).
3. Problem 2 proves that if the initial conditions φ and ψ are odd functions, then the solution u(x,t) is also an odd function.
Lesson 27: Integration by Substitution, part II (Section 10 version)Matthew Leingang
The document is the notes for a Calculus I class. It provides announcements for the upcoming class on Monday, which will involve reviewing course material rather than new topics. Examples are given of integration by substitution, including exponential, odd, and even functions. Multiple methods for substitutions are presented. The properties of odd and even functions are defined, and examples are shown graphically. Symmetric functions and their behavior under combinations are discussed.
Lesson 29: Integration by Substition (worksheet solutions)Matthew Leingang
This document contains the notes from a calculus class. It provides announcements about the final exam schedule and review sessions. It then discusses the technique of u-substitution for both indefinite and definite integrals. Examples are provided to illustrate how to use u-substitution to evaluate integrals involving trigonometric, polynomial, and other functions. The document emphasizes that u-substitution often makes evaluating integrals much easier than expanding them out directly.
This document is the outline for a calculus class. It discusses the final exam date and review sessions, and outlines the topics of substitution for indefinite integrals and substitution for definite integrals. It provides an example of using substitution to find the integral of the square root of x^2 + 1 by letting u = x^2 + 1, and expresses this using both standard notation and Leibniz notation. It states the theorem of substitution rule.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 27: Integration by Substitution (Section 041 slides)Matthew Leingang
The document contains notes from a Calculus I class at New York University on December 13, 2010. It discusses using the substitution method for indefinite and definite integrals. Examples are provided to demonstrate how to use substitutions to evaluate integrals involving trigonometric, exponential, and polynomial functions. The key steps are to make a substitution for the variable in terms of a new variable, determine the differential of the substitution, and substitute into the integral to transform it into an integral involving only the new variable.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
Lesson 27: Integration by Substitution (Section 4 version)Matthew Leingang
The document outlines a calculus lecture on integration by substitution. It provides examples of using u-substitution to find antiderivatives of expressions like √(x^2+1) and tan(x). The key ideas are that if u is a function of x, its derivative du/dx can be used to rewrite the integrand and perform a u-substitution integration.
1. The document provides solutions to homework problems involving partial differential equations.
2. Problem 1 solves the wave equation utt = c2uxx using d'Alembert's formula to find the solution u(x,t).
3. Problem 2 proves that if the initial conditions φ and ψ are odd functions, then the solution u(x,t) is also an odd function.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Lesson 9: The Product and Quotient Rules (slides)Matthew Leingang
The product rule is generally better because:
- It is more systematic and avoids mistakes from expanding products
- It works for any differentiable functions u and v, not just polynomials
- It provides insight into the structure of the derivative that direct computation does not
So in this example, using the product rule is preferable to direct multiplication.
The document provides examples demonstrating the chain rule for differentiating composite functions. The chain rule states that if y = g(u) and u = f(x), then dy/dx = (dy/du) * (du/dx). Several examples are worked through applying the chain rule to functions composed of multiple operations like sin(x^2) or (x^2 + 5x - 1)^(2/3). The chain rule can be extended to chains of more than two functions as shown in later examples.
This document provides an overview of engineering mathematics II with a focus on first order ordinary differential equations (ODEs). It explains what first order ODEs are, how to solve separable and reducible first order ODEs, and provides examples of applying first order ODEs to model real-world scenarios like population growth, decay, and radioactive decay. The objectives are to explain first order ODEs, separable equations, and apply the concepts to real life applications.
The document summarizes the Metropolis-adjusted Langevin algorithm (MALA) for sampling from log-concave probability measures in high dimensions. It introduces MALA and different proposal distributions, including random walk, Ornstein-Uhlenbeck, and Euler proposals. It discusses known results on optimal scaling, diffusion limits, ergodicity, and mixing time bounds. The main result is a contraction property for the MALA transition kernel under appropriate assumptions, implying dimension-independent bounds on mixing times.
This section introduces general and particular solutions to differential equations of the form y' = f(x) through direct integration and evaluation of constants. Examples provided include:
1) Integrating y' = 2x + 1 and applying the initial condition x = 0, y = 3 yields the general solution y(x) = x^2 + x + 3.
2) Integrating y' = (x - 2)^2 and applying x = 2, y = 1 yields y(x) = (1/3)(x - 2)^3.
3) Six more examples of first-order differential equations are worked through to find their general solutions.
The document discusses the idea of integration by parts, which involves using the product rule in reverse to evaluate integrals that cannot be solved using other methods. It presents the integration by parts formula as u(x)v'(x)dx = u(x)v(x) - u'(x)v(x)dx and works through an example problem of evaluating the integral of xex dx using this formula. The example breaks the integral down into separate terms and shows that the overall integral equals xex - ex + C.
Lesson 27: Integration by Substitution (Section 10 version)Matthew Leingang
The method of substitution is the chain rule in reverse. At first it looks magical, then logical, and then you realize there's an art to choosing the right substitution. We try to demystify with many worked-out examples.
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
This document discusses models of the atom and electron configuration. It begins by describing historical atomic models including Rutherford's model with a small, dense nucleus and electrons in orbits. Bohr's model improved on this by proposing electrons exist in specific energy levels. The modern quantum mechanical model describes electrons as probability clouds rather than definite orbits. The document then discusses electron configuration notation, including building up configurations using the aufbau principle and exceptions due to Hund's rule and half-filled orbitals. It concludes by introducing atomic spectra and the relationship between light and electron energy levels.
1. Cells are the basic units of structure and function in living organisms. A microscope is needed to view cells because they are too small to be seen with the naked eye.
2. The basic parts of a cell include a cell membrane, cytoplasm, and organelles. Eukaryotic cells also contain a nucleus. Cells come in two basic types - prokaryotic cells lack a nucleus while eukaryotic cells contain a nucleus.
3. Cells are organized into tissues, organs, and organ systems to carry out specialized functions in living organisms.
Biology - Chp 1 - Biology The Study Of Life - PowerPointMel Anthony Pepito
1. The document provides instructions for students to complete a "Do Now" assignment at the beginning of class involving defining what makes something alive.
2. It then summarizes key concepts from Chapter 1 of a biology textbook including defining biology as the study of life and outlining characteristics of living things such as organization, reproduction, growth/development, response to stimuli, and evolution.
3. The methods used in biology like observation, hypothesis formation, experimentation, data analysis and theory development are described along with examples. Safety procedures and metric units are also covered.
1. The document discusses covalent bonding and molecular compounds. It defines covalent bonds as the sharing of electrons between nonmetal atoms.
2. Molecular compounds are formed from covalent bonds between atoms. They have lower melting and boiling points than ionic compounds.
3. Molecular formulas show the number and type of atoms in a molecule, but not their arrangement. Water's molecular formula is H2O.
This document contains notes from a chemistry chapter about matter and changes. It defines matter as anything that has mass and volume. It describes properties of matter as either extensive, depending on amount, or intensive, depending on type. The three states of matter are solids, liquids, and gases. Physical changes alter a material's properties without changing its composition and can be classified as reversible or irreversible. Mixtures contain two or more substances mixed together either homogeneously or heterogeneously. Solutions are homogeneous mixtures that can be separated using physical means like decanting, filtration, distillation, magnets, or sorting.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Lesson 9: The Product and Quotient Rules (slides)Matthew Leingang
The product rule is generally better because:
- It is more systematic and avoids mistakes from expanding products
- It works for any differentiable functions u and v, not just polynomials
- It provides insight into the structure of the derivative that direct computation does not
So in this example, using the product rule is preferable to direct multiplication.
The document provides examples demonstrating the chain rule for differentiating composite functions. The chain rule states that if y = g(u) and u = f(x), then dy/dx = (dy/du) * (du/dx). Several examples are worked through applying the chain rule to functions composed of multiple operations like sin(x^2) or (x^2 + 5x - 1)^(2/3). The chain rule can be extended to chains of more than two functions as shown in later examples.
This document provides an overview of engineering mathematics II with a focus on first order ordinary differential equations (ODEs). It explains what first order ODEs are, how to solve separable and reducible first order ODEs, and provides examples of applying first order ODEs to model real-world scenarios like population growth, decay, and radioactive decay. The objectives are to explain first order ODEs, separable equations, and apply the concepts to real life applications.
The document summarizes the Metropolis-adjusted Langevin algorithm (MALA) for sampling from log-concave probability measures in high dimensions. It introduces MALA and different proposal distributions, including random walk, Ornstein-Uhlenbeck, and Euler proposals. It discusses known results on optimal scaling, diffusion limits, ergodicity, and mixing time bounds. The main result is a contraction property for the MALA transition kernel under appropriate assumptions, implying dimension-independent bounds on mixing times.
This section introduces general and particular solutions to differential equations of the form y' = f(x) through direct integration and evaluation of constants. Examples provided include:
1) Integrating y' = 2x + 1 and applying the initial condition x = 0, y = 3 yields the general solution y(x) = x^2 + x + 3.
2) Integrating y' = (x - 2)^2 and applying x = 2, y = 1 yields y(x) = (1/3)(x - 2)^3.
3) Six more examples of first-order differential equations are worked through to find their general solutions.
The document discusses the idea of integration by parts, which involves using the product rule in reverse to evaluate integrals that cannot be solved using other methods. It presents the integration by parts formula as u(x)v'(x)dx = u(x)v(x) - u'(x)v(x)dx and works through an example problem of evaluating the integral of xex dx using this formula. The example breaks the integral down into separate terms and shows that the overall integral equals xex - ex + C.
Lesson 27: Integration by Substitution (Section 10 version)Matthew Leingang
The method of substitution is the chain rule in reverse. At first it looks magical, then logical, and then you realize there's an art to choosing the right substitution. We try to demystify with many worked-out examples.
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
This document discusses models of the atom and electron configuration. It begins by describing historical atomic models including Rutherford's model with a small, dense nucleus and electrons in orbits. Bohr's model improved on this by proposing electrons exist in specific energy levels. The modern quantum mechanical model describes electrons as probability clouds rather than definite orbits. The document then discusses electron configuration notation, including building up configurations using the aufbau principle and exceptions due to Hund's rule and half-filled orbitals. It concludes by introducing atomic spectra and the relationship between light and electron energy levels.
1. Cells are the basic units of structure and function in living organisms. A microscope is needed to view cells because they are too small to be seen with the naked eye.
2. The basic parts of a cell include a cell membrane, cytoplasm, and organelles. Eukaryotic cells also contain a nucleus. Cells come in two basic types - prokaryotic cells lack a nucleus while eukaryotic cells contain a nucleus.
3. Cells are organized into tissues, organs, and organ systems to carry out specialized functions in living organisms.
Biology - Chp 1 - Biology The Study Of Life - PowerPointMel Anthony Pepito
1. The document provides instructions for students to complete a "Do Now" assignment at the beginning of class involving defining what makes something alive.
2. It then summarizes key concepts from Chapter 1 of a biology textbook including defining biology as the study of life and outlining characteristics of living things such as organization, reproduction, growth/development, response to stimuli, and evolution.
3. The methods used in biology like observation, hypothesis formation, experimentation, data analysis and theory development are described along with examples. Safety procedures and metric units are also covered.
1. The document discusses covalent bonding and molecular compounds. It defines covalent bonds as the sharing of electrons between nonmetal atoms.
2. Molecular compounds are formed from covalent bonds between atoms. They have lower melting and boiling points than ionic compounds.
3. Molecular formulas show the number and type of atoms in a molecule, but not their arrangement. Water's molecular formula is H2O.
This document contains notes from a chemistry chapter about matter and changes. It defines matter as anything that has mass and volume. It describes properties of matter as either extensive, depending on amount, or intensive, depending on type. The three states of matter are solids, liquids, and gases. Physical changes alter a material's properties without changing its composition and can be classified as reversible or irreversible. Mixtures contain two or more substances mixed together either homogeneously or heterogeneously. Solutions are homogeneous mixtures that can be separated using physical means like decanting, filtration, distillation, magnets, or sorting.
This document provides an overview of key concepts about matter and chemical changes from a chemistry textbook. It defines matter and its three main states (solid, liquid, gas). It describes properties as either extensive (depending on amount) or intensive (depending on type). It differentiates between physical and chemical changes, elements and compounds, and mixtures and pure substances. It outlines clues that indicate a chemical change has occurred and introduces the law of conservation of mass.
The document defines mixed numbers and improper fractions. A mixed number has two parts - a whole number and a fraction, while an improper fraction has a numerator larger than the denominator. The document provides steps for changing between mixed numbers and improper fractions, including multiplying the denominator by the whole number and adding it to the numerator.
This document discusses properties of solutions and concentration of solutions. It covers factors that determine the rate and amount of solute that dissolves in a solvent, such as temperature, stirring, and surface area. The maximum amount of solute that can dissolve is called the solubility. Units used to express solubility include grams of solute per 100 grams of solvent. The concentration of a solution can be quantified using molarity, which is the moles of solute per liter of solution. More concentrated solutions have more solute dissolved in a smaller amount of solvent, while dilute solutions have less solute in a larger amount of solvent.
The document discusses scientific measurement and units. It covers accuracy, precision, and significant figures when making measurements. Conversion factors allow measurements to be converted between different units through multiplication. Dimensional analysis uses the units of measurements to solve conversion problems by breaking them into steps. Complex problems are best solved by breaking them into manageable parts.
Fungi are eukaryotic heterotrophs that feed by absorbing nutrients from outside their bodies. They have cell walls containing chitin and their bodies are composed of filaments called hyphae that form a tangled mass called a mycelium. Fungi reproduce both sexually and asexually through spores. They play important ecological roles as decomposers that break down organic matter, parasites that infect plants and animals, and symbionts that form relationships like lichens and mycorrhizae.
Chemistry - Chp 1 - Introduction To Chemistry - PowerPointMel Anthony Pepito
The document introduces chemistry by defining it as the study of matter, its properties, and the changes it undergoes, discussing the major areas of chemistry and how it relates to everyday life and future careers, and outlining the scientific method and approaches to solving problems, emphasizing developing a plan, performing calculations correctly for numeric problems, and applying concepts for conceptual problems.
The document discusses techniques for sketching graphs of functions, including:
- Using the increasing/decreasing test to determine if a function is increasing or decreasing based on the sign of the derivative
- Using the concavity test to determine if a graph is concave up or down based on the second derivative
- A checklist for completely graphing a function, including finding critical points, inflection points, asymptotes, and putting together the information about monotonicity and concavity.
The document summarizes key concepts from a chapter on the history of life as revealed by the fossil record. It describes how paleontologists study fossils to understand past life forms and environments, and how fossils provide evidence that life has changed over time with most species going extinct. It also discusses techniques for dating fossils, important patterns in macroevolution like extinction events, adaptive radiation and convergent evolution, and the role of developmental genes in shaping body plans over long periods of time.
Chemistry - Chp 6 - The Periodic Table Revisited - PowerPointMel Anthony Pepito
The document discusses how elements are organized in the periodic table based on their atomic structure and properties, including trends in atomic size, ion formation, and other periodic properties as the atomic number increases and across periods. Early periodic tables organized just a few elements, but the modern periodic table is based on the periodic law and arranges all known elements in order of increasing atomic number and identifies trends in their physical and chemical properties.
The document contains a series of questions about identifying synonyms. It presents a word in bold and then 3 potential synonyms as answer choices to choose from. Some of the questions include identifying synonyms for words like "doubtful", "annual", "mandatory", "occasion", and "insolent". The purpose is to test the reader's knowledge of synonyms and ability to choose the word that means the same thing as the given word.
The document provides steps for evaluating expressions with variables. It works through examples of evaluating 6x, 6x3, 5n-3 for different values of n, 3.6/y + 2.8 for different values of y. For each example it shows substituting the value for the variable, then performing the calculations step-by-step according to the order of operations, providing the final solution.
Chemistry - Chp 7 - Ionic and Metallic Bonding - PowerPointMel Anthony Pepito
This document summarizes key concepts from Chapter 7 on ionic and metallic bonding. It discusses how ions form via gaining or losing valence electrons to achieve stable noble gas configurations, and how ionic and metallic bonds differ. Ionic compounds are crystalline solids with high melting points that conduct when molten or dissolved. Metals have mobile valence electrons that allow conduction and properties like malleability. Alloys combine metals for improved properties.
Biology - Chp 2 - Hydrolysis And Dehydration Synthesis - PowerPointMel Anthony Pepito
Trypsin works best at basic pH levels while pepsin works best at acidic pH levels found in the stomach. Raising the stomach's pH would decrease pepsin's effectiveness at breaking down proteins. The document discusses how hydrolysis breaks down complex molecules like maltose into simpler molecules like glucose through the addition of water. Dehydration synthesis combines simpler molecules like glucose together to form more complex ones like maltose or starch by removing water.
This document provides an overview of ecology and key ecological concepts. It discusses the biosphere and levels of ecological organization from species to biomes. The document covers ecological methods of observation, experimentation and modeling. It then examines energy flow through ecosystems, including producers, consumers, trophic levels, and ecological pyramids. Finally, the document details the water, carbon, nitrogen, phosphorus and other nutrient cycles that allow for recycling of matter within ecosystems. Nutrient limitations can impact an ecosystem's primary productivity.
This chapter discusses methods for measuring quantities of substances. It introduces the mole as a unit for measuring amounts of matter equal to Avogadro's number of representative particles. The mole allows for easy conversion between mass and number of particles. Compound formulas allow calculating formula mass in grams per mole. Relationships between moles, mass, and volume at standard temperature and pressure enable interconversion between these quantities.
Lesson 27: Integration by Substitution (Section 041 slides)Mel Anthony Pepito
The document provides notes from a Calculus I class at New York University on December 13, 2010. It includes announcements about upcoming review sessions and the final exam. It then outlines the objectives and topics to be covered, which is integration by substitution. Examples are provided to demonstrate how to use u-substitution to evaluate indefinite integrals. The key steps are to let u equal an expression involving x, find du in terms of dx, and then substitute into the integral.
This document discusses differentiating functions involving logarithmic and exponential terms. It covers:
1) The derivatives of ln(x) and e^x as 1/x and e^x, respectively.
2) Using product, quotient, and chain rules to differentiate functions containing ln(x) or e^x, including examples of differentiating expressions with both ln(x) and e^x terms.
3) A caution about confusing the power rule and exponential functions when taking derivatives.
The document provides examples of differentiating various logarithmic and exponential expressions and encourages practicing differentiating expressions involving both ln(x) and e^x.
This document discusses implicit differentiation, which is a technique for taking the derivative of equations that cannot be solved explicitly for y as a function of x. It explains that when differentiating terms involving both x and y, the derivative of the y term is dy/dx. As an example, it shows the differentiation of xy using the product rule, which yields y + x*dy/dx. The document concludes by applying this technique to differentiate the equation y4 + xy = x3 - x + 2 implicitly with respect to x.
1) The document discusses integration as the reverse process of differentiation, and provides examples of integrating logarithmic and exponential functions.
2) Key points covered include integrating 1/x as ln(x), and exponential functions like e3x by using u-substitution and dividing the "tail" by the exponent.
3) The document also demonstrates integrating logarithmic expressions using u-substitution, letting the term in the denominator equal u and finding du.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
This document contains the answers to exercises for the third edition of the textbook "Microeconomic Analysis" by Hal R. Varian. The answers are organized by chapter and include solutions to mathematical problems as well as explanations and justifications. Key information provided in the answers includes derivations of production functions, profit functions, cost functions, and factor demand functions for various technologies. Convexity and monotonicity properties of technologies are also analyzed.
The document provides examples of derivatives and their corresponding anti-derivatives (indefinite integrals) for various functions. It also demonstrates rules for taking the anti-derivative of functions using u-substitution. Some key rules covered include adding +c to account for constants and applying power rules for integrals involving terms like 4x, 3x^2, or other polynomial functions. Examples are worked through step-by-step to illustrate properly applying u-substitution and integrating more complex expressions.
Trigonometric substitutions can help evaluate integrals involving trigonometric functions. The document provides examples of using trigonometric substitutions to evaluate integrals of powers of sine and cosine functions. Specifically:
1) Integrals of powers of sine and cosine can be evaluated by rewriting the functions using trigonometric identities and then substituting variables. For example, sin5x can be rewritten as sinx(sin2x)2 and evaluated using the substitution u=cosx.
2) More complex integrals may require multiple steps and identities. For example, evaluating sin6x requires rewriting sin2x using an identity and then integrating four resulting terms.
3) Trigonometric substitutions allow rewriting integrals involving
Lesson 26: The Fundamental Theorem of Calculus (slides)Mel Anthony Pepito
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the anti-derivative F of f. Examples are provided to illustrate how to use the Fundamental Theorem to find derivatives and integrals.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
The document discusses various techniques for finding antiderivatives (indefinite integrals). It covers:
1) Using the power rule to find antiderivatives by increasing the exponent by 1 and dividing by the new exponent.
2) Rewriting expressions with rational or negative exponents before taking the antiderivative.
3) Expanding expressions, like using FOIL, before taking the antiderivative term by term.
4) Simplifying expressions, like factoring, before taking the antiderivative.
5) Setting up and solving word problems involving antiderivatives to find functions for quantities like profit, distance, and rate of change.
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3) Using a substitution to transform a differential equation into an exact form and finding its solution.
4) Finding the velocity as a function of time by solving an initial value problem.
5) Solving non-homogeneous linear differential equations with constant coefficients by using the method of undetermined coefficients.
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1) It discusses the derivatives of exponential functions with any base, as well as the derivatives of logarithmic functions with any base.
2) It covers using the technique of logarithmic differentiation to find derivatives of functions involving products, quotients, and/or exponentials.
3) The document provides examples of finding derivatives of various logarithmic and exponential functions.
The product rule can be iterated to find the derivative of products with more than two factors. The derivative of a three-factor product uvw is u'vw + uv'w + uvw'. More generally, the derivative of a product of n factors breaks the product into a sum of n terms by applying the product rule recursively.
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2) Expanding rational or negative exponents before integrating
3) Expanding expressions before integrating term by term
4) Simplifying rational expressions by factoring and canceling before integrating
5) Setting up word problems involving integrals to find related functions like total cost, revenue, distance over time.
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Similar to Lesson 27: Integration by Substitution (slides) (20)
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Lesson 27: Integration by Substitution (slides)
1. Sec on 5.5
Integra on by Subs tu on
V63.0121.011: Calculus I
Professor Ma hew Leingang
New York University
May 4, 2011
.
2. Announcements
Today: 5.5
Monday 5/9: Review in class
Tuesday 5/10: Review Sessions by TAs
Wednesday 5/11: TA office hours:
Adam 10–noon (WWH 906)
Jerome 3:30–5:30 (WWH 501)
Soohoon 6–8 (WWH 511)
Thursday 5/12: Final Exam,
2:00–3:50pm, CANT 200
3. Resurrection Policy
If your final score beats your midterm score, we will add 10% to its
weight, and subtract 10% from the midterm weight.
Image credit: Sco Beale / Laughing Squid
4. Objectives
Given an integral and a
subs tu on, transform the
integral into an equivalent one
using a subs tu on
Evaluate indefinite integrals
using the method of
subs tu on.
Evaluate definite integrals using
the method of subs tu on.
5. Outline
Last Time: The Fundamental Theorem(s) of Calculus
Subs tu on for Indefinite Integrals
Theory
Examples
Subs tu on for Definite Integrals
Theory
Examples
6. Differentiation and Integration as
reverse processes
Theorem (The Fundamental Theorem of Calculus)
1. Let f be con nuous on [a, b]. Then
∫
d x
f(t) dt = f(x)
dx a
2. Let f be con nuous on [a, b] and f = F′ for some other func on
F. Then ∫ b
f(x) dx = F(b) − F(a).
a
7. Techniques of antidifferentiation?
So far we know only a few rules for an differen a on. Some are
general, like
∫ ∫ ∫
[f(x) + g(x)] dx = f(x) dx + g(x) dx
8. Techniques of antidifferentiation?
So far we know only a few rules for an differen a on. Some are
general, like
∫ ∫ ∫
[f(x) + g(x)] dx = f(x) dx + g(x) dx
Some are pre y par cular, like
∫
1
√ dx = arcsec x + C.
x x2 − 1
9. Techniques of antidifferentiation?
So far we know only a few rules for an differen a on. Some are
general, like
∫ ∫ ∫
[f(x) + g(x)] dx = f(x) dx + g(x) dx
Some are pre y par cular, like
∫
1
√ dx = arcsec x + C.
x x2 − 1
What are we supposed to do with that?
10. No straightforward system of
antidifferentiation
So far we don’t have any way to find
∫
2x
√ dx
x2 + 1
or ∫
tan x dx.
11. No straightforward system of
antidifferentiation
So far we don’t have any way to find
∫
2x
√ dx
x2 + 1
or ∫
tan x dx.
Luckily, we can be smart and use the “an ” version of one of the
most important rules of differen a on: the chain rule.
12. Outline
Last Time: The Fundamental Theorem(s) of Calculus
Subs tu on for Indefinite Integrals
Theory
Examples
Subs tu on for Definite Integrals
Theory
Examples
14. Substitution for Indefinite
Integrals
Example
Find ∫
x
√ dx.
x2 + 1
Solu on
Stare at this long enough and you no ce the the integrand is the
√
deriva ve of the expression 1 + x2 .
19. Leibnizian notation FTW
Solu on (Same technique, new nota on)
√ √
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u.
20. Leibnizian notation FTW
Solu on (Same technique, new nota on)
√ √
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. So the
integrand becomes completely transformed into
∫ ∫ 1 ∫
x dx 2 du 1
√ = √ = √ du
x2 + 1 u 2 u
21. Leibnizian notation FTW
Solu on (Same technique, new nota on)
√ √
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. So the
integrand becomes completely transformed into
∫ ∫ 1 ∫
x dx 2 du 1
√ = √ = √ du
x2 + 1 ∫ u 2 u
1 −1/2
= 2u du
22. Leibnizian notation FTW
Solu on (Same technique, new nota on)
√ √
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. So the
integrand becomes completely transformed into
∫ ∫ 1 ∫
x dx 2 du 1
√ = √ = √ du
x2 + 1 ∫ u 2 u
1 −1/2
= 2u du
√ √
= u + C = 1 + x2 + C.
23. Useful but unsavory variation
Solu on (Same technique, new nota on, more idiot-proof)
√ √
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. “Solve for dx:”
du
dx =
2x
24. Useful but unsavory variation
Solu on (Same technique, new nota on, more idiot-proof)
√ √
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. “Solve for dx:”
du
dx =
2x
So the integrand becomes completely transformed into
∫ ∫
x x du
√ dx = √ ·
x2 + 1 u 2x
25. Useful but unsavory variation
Solu on (Same technique, new nota on, more idiot-proof)
√ √
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. “Solve for dx:”
du
dx =
2x
So the integrand becomes completely transformed into
∫ ∫ ∫
x x du 1
√ dx = √ · = √ du
x2 + 1 u 2x 2 u
26. Useful but unsavory variation
Solu on (Same technique, new nota on, more idiot-proof)
√ √
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. “Solve for dx:”
du
dx =
2x
So the integrand becomes completely transformed into
∫ ∫ ∫
x x du 1
√ dx = √ · = √ du
x2 + 1 u 2x 2 u
∫
1 −1/2
= 2u du
27. Useful but unsavory variation
Solu on (Same technique, new nota on, more idiot-proof)
√ √
Let u = x2 + 1. Then du = 2x dx and 1 + x2 = u. “Solve for dx:”
du
dx =
2x
So the integrand becomes completely transformed into
∫ ∫ ∫
x x du 1
√ dx = √ · = √ du
x2 + 1 u 2x 2 u
∫ √
1 −1/2
√
= 2 u du = u + C = 1 + x2 + C.
28. Theorem of the Day
Theorem (The Subs tu on Rule)
If u = g(x) is a differen able func on whose range is an interval I
and f is con nuous on I, then
∫ ∫
′
f(g(x))g (x) dx = f(u) du
29. Theorem of the Day
Theorem (The Subs tu on Rule)
If u = g(x) is a differen able func on whose range is an interval I
and f is con nuous on I, then
∫ ∫
′
f(g(x))g (x) dx = f(u) du
That is, if F is an an deriva ve for f, then
∫
f(g(x))g′ (x) dx = F(g(x))
30. Theorem of the Day
Theorem (The Subs tu on Rule)
If u = g(x) is a differen able func on whose range is an interval I
and f is con nuous on I, then
∫ ∫
′
f(g(x))g (x) dx = f(u) du
In Leibniz nota on:
∫ ∫
du
f(u) dx = f(u) du
dx
31. A polynomial example
Example
∫
Use the subs tu on u = x2 + 3 to find (x2 + 3)3 4x dx.
32. A polynomial example
Example
∫
Use the subs tu on u = x2 + 3 to find (x2 + 3)3 4x dx.
Solu on
If u = x2 + 3, then du = 2x dx, and 4x dx = 2 du. So
∫
(x2 + 3)3 4x dx
33. A polynomial example
Example
∫
Use the subs tu on u = x2 + 3 to find (x2 + 3)3 4x dx.
Solu on
If u = x2 + 3, then du = 2x dx, and 4x dx = 2 du. So
∫ ∫ ∫
(x + 3) 4x dx = u 2du = 2 u3 du
2 3 3
34. A polynomial example
Example
∫
Use the subs tu on u = x2 + 3 to find (x2 + 3)3 4x dx.
Solu on
If u = x2 + 3, then du = 2x dx, and 4x dx = 2 du. So
∫ ∫ ∫
(x + 3) 4x dx = u 2du = 2 u3 du
2 3 3
1
= u4
2
35. A polynomial example
Example
∫
Use the subs tu on u = x2 + 3 to find (x2 + 3)3 4x dx.
Solu on
If u = x2 + 3, then du = 2x dx, and 4x dx = 2 du. So
∫ ∫ ∫
(x + 3) 4x dx = u 2du = 2 u3 du
2 3 3
1 1
= u4 = (x2 + 3)4
2 2
41. A polynomial example (brute force)
Solu on
∫ ∫
2 3
( )
(x + 3) 4x dx = x6 + 9x4 + 27x2 + 27 4x dx
∫
( )
= 4x7 + 36x5 + 108x3 + 108x dx
1
= x8 + 6x6 + 27x4 + 54x2
2
Which would you rather do?
42. A polynomial example (brute force)
Solu on
∫ ∫
2 3
( )
(x + 3) 4x dx = x6 + 9x4 + 27x2 + 27 4x dx
∫
( )
= 4x7 + 36x5 + 108x3 + 108x dx
1
= x8 + 6x6 + 27x4 + 54x2
2
Which would you rather do?
It’s a wash for low powers
43. A polynomial example (brute force)
Solu on
∫ ∫
2 3
( )
(x + 3) 4x dx = x6 + 9x4 + 27x2 + 27 4x dx
∫
( )
= 4x7 + 36x5 + 108x3 + 108x dx
1
= x8 + 6x6 + 27x4 + 54x2
2
Which would you rather do?
It’s a wash for low powers
But for higher powers, it’s much easier to do subs tu on.
44. Compare
We have the subs tu on method, which, when mul plied out, gives
∫
1
(x2 + 3)3 4x dx = (x2 + 3)4
2
1( 8 )
= x + 12x6 + 54x4 + 108x2 + 81
2
1 81
= x8 + 6x6 + 27x4 + 54x2 +
2 2
and the brute force method
∫
1
(x2 + 3)3 4x dx = x8 + 6x6 + 27x4 + 54x2
2
Is there a difference? Is this a problem?
45. Compare
We have the subs tu on method, which, when mul plied out, gives
∫
1
(x2 + 3)3 4x dx = (x2 + 3)4 + C
2
1( 8 )
= x + 12x6 + 54x4 + 108x2 + 81 + C
2
1 81
= x8 + 6x6 + 27x4 + 54x2 + +C
2 2
and the brute force method
∫
1
(x2 + 3)3 4x dx = x8 + 6x6 + 27x4 + 54x2 + C
2
Is there a difference? Is this a problem? No, that’s what +C means!
47. A slick example
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
48. A slick example
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solu on
Let u = cos x . Then du = − sin x dx . So
∫ ∫
sin x
tan x dx = dx
cos x
49. A slick example
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solu on
Let u = cos x . Then du = − sin x dx . So
∫ ∫
sin x
tan x dx = dx
cos x
50. A slick example
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solu on
Let u = cos x . Then du = − sin x dx . So
∫ ∫
sin x
tan x dx = dx
cos x
51. A slick example
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solu on
Let u = cos x . Then du = − sin x dx . So
∫ ∫ ∫
sin x 1
tan x dx = dx = − du
cos x u
52. A slick example
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solu on
Let u = cos x . Then du = − sin x dx . So
∫ ∫ ∫
sin x 1
tan x dx = dx = − du
cos x u
= − ln |u| + C
53. A slick example
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solu on
Let u = cos x . Then du = − sin x dx . So
∫ ∫ ∫
sin x 1
tan x dx = dx = − du
cos x u
= − ln |u| + C
= − ln | cos x| + C = ln | sec x| + C
54. Can you do it another way?
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
55. Can you do it another way?
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solu on
du
Let u = sin x. Then du = cos x dx and so dx = .
cos x
56. Can you do it another way?
Example
∫
sin x
Find tan x dx. (Hint: tan x = )
cos x
Solu on
du
Let u = sin x. Then du = cos x dx and so dx = .
cos x
∫ ∫ ∫
sin x u du
tan x dx = dx =
∫ cos x ∫ cos x cos x ∫
u du u du u du
= = =
cos2 x 1 − sin2 x 1 − u2
57. For those who really must know all
Solu on (Con nued, with algebra help)
Let y = 1 − u2 , so dy = −2u du. Then
∫ ∫ ∫
u du u dy
tan x dx = =
1∫− u2 y −2u
1 dy 1 1
=− = − ln |y| + C = − ln 1 − u2 + C
2 y 2 2
1 1
= ln √ + C = ln √ +C
1 − u2 1 − sin2 x
1
= ln + C = ln |sec x| + C
|cos x|
58. Outline
Last Time: The Fundamental Theorem(s) of Calculus
Subs tu on for Indefinite Integrals
Theory
Examples
Subs tu on for Definite Integrals
Theory
Examples
59. Substitution for Definite Integrals
Theorem (The Subs tu on Rule for Definite Integrals)
If g′ is con nuous and f is con nuous on the range of u = g(x), then
∫ b ∫ g(b)
′
f(g(x))g (x) dx = f(u) du.
a g(a)
60. Substitution for Definite Integrals
Theorem (The Subs tu on Rule for Definite Integrals)
If g′ is con nuous and f is con nuous on the range of u = g(x), then
∫ b ∫ g(b)
′
f(g(x))g (x) dx = f(u) du.
a g(a)
Why the change in the limits?
The integral on the le happens in “x-land”
The integral on the right happens in “u-land”, so the limits need
to be u-values
To get from x to u, apply g
62. Example
∫ π
Compute cos2 x sin x dx.
0
Solu on (Slow Way)
63. Example
∫ π
Compute cos2 x sin x dx.
0
Solu on (Slow Way) ∫
First compute the indefinite integral cos2 x sin x dx and then
evaluate.
Let u = cos x . Then du = − sin x dx and
∫
cos2 x sin x dx
64. Example
∫ π
Compute cos2 x sin x dx.
0
Solu on (Slow Way) ∫
First compute the indefinite integral cos2 x sin x dx and then
evaluate.
Let u = cos x . Then du = − sin x dx and
∫
cos2 x sin x dx
65. Example
∫ π
Compute cos2 x sin x dx.
0
Solu on (Slow Way) ∫
First compute the indefinite integral cos2 x sin x dx and then
evaluate.
Let u = cos x . Then du = − sin x dx and
∫
cos2 x sin x dx
66. Example
∫ π
Compute cos2 x sin x dx.
0
Solu on (Slow Way) ∫
First compute the indefinite integral cos2 x sin x dx and then
evaluate.
Let u = cos x . Then du = − sin x dx and
∫ ∫
cos2 x sin x dx = − u2 du
67. Example
∫ π
Compute cos2 x sin x dx.
0
Solu on (Slow Way) ∫
First compute the indefinite integral cos2 x sin x dx and then
evaluate.
Let u = cos x . Then du = − sin x dx and
∫ ∫
cos2 x sin x dx = − u2 du = − 1 u3 + C = − 3 cos3 x + C.
3
1
68. Example
∫ π
Compute cos2 x sin x dx.
0
Solu on (Slow Way)
Let u = cos x . Then du = − sin x dx and
∫ ∫
cos x sin x dx = − u2 du = − 1 u3 + C = − 3 cos3 x + C.
2
3
1
Therefore
∫ π π
1
cos2 x sin x dx = − cos3 x
0 3 0
69. Example
∫ π
Compute cos2 x sin x dx.
0
Solu on (Slow Way)
Let u = cos x . Then du = − sin x dx and
∫ ∫
cos x sin x dx = − u2 du = − 1 u3 + C = − 3 cos3 x + C.
2
3
1
Therefore
∫ π
1( )
π
1
cos2 x sin x dx = − cos3 x =− (−1)3 − 13
0 3 0 3
70. Example
∫ π
Compute cos2 x sin x dx.
0
Solu on (Slow Way)
Let u = cos x . Then du = − sin x dx and
∫ ∫
cos x sin x dx = − u2 du = − 1 u3 + C = − 3 cos3 x + C.
2
3
1
Therefore
∫ π
1( ) 2
π
1
cos2 x sin x dx = − cos3 x =− (−1)3 − 13 = .
0 3 0 3 3
71. Definite-ly Quicker
Solu on (Fast Way)
Do both the subs tu on and the evalua on at the same me. Let
u = cos x. Then du = − sin x dx, u(0) = 1 and u(π) = −1 . So
∫ π
cos2 x sin x dx
0
72. Definite-ly Quicker
Solu on (Fast Way)
Do both the subs tu on and the evalua on at the same me. Let
u = cos x. Then du = − sin x dx, u(0) = 1 and u(π) = −1 . So
∫ π
cos2 x sin x dx
0
73. Definite-ly Quicker
Solu on (Fast Way)
Do both the subs tu on and the evalua on at the same me. Let
u = cos x. Then du = − sin x dx, u(0) = 1 and u(π) = −1 . So
∫ π
cos2 x sin x dx
0
74. Definite-ly Quicker
Solu on (Fast Way)
Do both the subs tu on and the evalua on at the same me. Let
u = cos x. Then du = − sin x dx, u(0) = 1 and u(π) = −1 . So
∫ π ∫ −1 ∫ 1
2
cos x sin x dx = −u du =
2
u2 du
0 1 −1
75. Definite-ly Quicker
Solu on (Fast Way)
Do both the subs tu on and the evalua on at the same me. Let
u = cos x. Then du = − sin x dx, u(0) = 1 and u(π) = −1 . So
∫ π ∫ −1 ∫ 1
2
cos x sin x dx = −u du =
2
u2 du
0 1 −1
1( ) 2
1
1 3
= u = 1 − (−1) =
3 −1 3 3
76. Compare
The advantage to the “fast way” is that you completely
transform the integral into something simpler and don’t have
to go back to the original variable (x).
But the slow way is just as reliable.
78. An exponential example
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
Let u = e2x , so du = 2e2x dx. We have
∫ √ ∫
ln 8
2x
√ 1 8√
√ e e2x + 1 dx = u + 1 du
ln 3 2 3
79. About those limits
Since
√ √ 2
e2(ln 3)
= eln 3
= eln 3 = 3
we have ∫ √ ∫
ln 8 √ 1 8√
√ e2x e2x + 1 dx = u + 1 du
ln 3 2 3
80. An exponential example
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
Now let y = u + 1, dy = du. So
∫ ∫
1 8√
9
1 9√ 1 2 1 19
u + 1 du = y dy = · y3/2 = (27 − 8) =
2 3 2 4 2 3 4 3 3
81. About those fractional powers
We have
93/2 = (91/2 )3 = 33 = 27
43/2 = (41/2 )3 = 23 = 8
so ∫ 9 9
1 1 2 1 19
1/2
y dy = · y3/2 = (27 − 8) =
2 4 2 3 4 3 3
82. An exponential example
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
Now let y = u + 1, dy = du. So
∫ ∫
1 8√
9
1 9√ 1 2 1 19
u + 1 du = y dy = · y3/2 = (27 − 8) =
2 3 2 4 2 3 4 3 3
83. Another way to skin that cat
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
Let u = e2x + 1,
84. Another way to skin that cat
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
Let u = e2x + 1,so that du = 2e2x dx.
85. Another way to skin that cat
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
Let u = e2x + 1,so that du = 2e2x dx. Then
∫ √ ∫
ln 8 √ 1 9√
√ e2x e2x + 1 dx = u du
ln 3 2 4
86. Another way to skin that cat
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
Let u = e2x + 1,so that du = 2e2x dx. Then
∫ √ ∫
ln 8 √ 1 9√ 1
9
√ e2x e2x + 1 dx = u du = u3/2
ln 3 2 4 3 4
87. Another way to skin that cat
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
Let u = e2x + 1,so that du = 2e2x dx. Then
∫ √ ∫
ln 8 √ 1 9√ 1
9
1 19
√ e2x e2x + 1 dx = u du = u3/2 = (27 − 8) =
ln 3 2 4 3 4 3 3
88. A third skinned cat
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
√
Let u = e2x + 1, so that u2 = e2x + 1
89. A third skinned cat
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
√
Let u = e2x + 1, so that u2 = e2x + 1 =⇒ 2u du = 2e2x dx
90. A third skinned cat
Example
∫ ln √8 √
Find √ e2x e2x + 1 dx
ln 3
Solu on
√
Let u = e2x + 1, so that u2 = e2x + 1 =⇒ 2u du = 2e2x dx Thus
∫ √ ∫ 3
ln 8 √ 1
3
19
√ e2x e2x + 1 dx = u · u du = u3 =
ln 3 2 3 2 3
92. A Trigonometric Example
Example
Find ∫ ( ) ( )
3π/2
θ θ
cot5 sec2 dθ.
π 6 6
Before we dive in, think about:
What “easy” subs tu ons might help?
Which of the trig func ons suggests a subs tu on?
98. Graphs
∫ 3π/2 ( ) ( ) ∫ π/4
θ θ
cot5 sec2 dθ 6 cot5 φ sec2 φ dφ
π 6 6 π/6
y y
. θ φ
π 3π ππ
2 64
The areas of these two regions are the same.
99. ∫ ∫
Graphs π/4
5 2
6 cot φ sec φ dφ
1
√ 6u
−5
du
π/6 1/ 3
y y
. φ u
ππ 11
√
64 3
The areas of these two regions are the same.
100. u/du pairs
When deciding on a subs tu on, look for sub-expressions where
one is (a constant mul ple of) the deriva ve of the other. Such as:
√
u xn ln x sin x cos x tan x x ex
1 1
constant × du xn−1 cos x sin x sec2 x √ ex
x x
101. Summary
If F is an an deriva ve for f, then:
∫
f(g(x))g′ (x) dx = F(g(x))
If F is an an deriva ve for f, which is con nuous on the range
of g, then:
∫ b ∫ g(b)
′
f(g(x))g (x) dx = f(u) du = F(g(b)) − F(g(a))
a g(a)
An differen a on in general and subs tu on in par cular is a
“nonlinear” problem that needs prac ce, intui on, and
perserverance.
The whole an differen a on story is in Chapter 6.