integration in maths pdf mathematics integration

1. INTEGRATION
‫التكامل‬

2. 1
Integration
Overview:
Integration is the calculation of an integral. Integrals in maths are used to
find many useful quantities such as areas, volumes, displacement, etc.
When we speak about integrals, it is related to usually definite integrals.
The indefinite integrals are used for antiderivatives. Integration is one of
the two major calculus topics in Mathematics, apart from differentiation
(which measure the rate of change of any function with respect to its
variables).
Integration is used in dealing with two essentially different types of
problems: The first types are problems in which the derivative of a
function, or its rate of change, or the slope of its graph, is known and we
want to find the function. We are therefore required to reverse the process
of differentiation. This reverse process is known as antidifferentiation, or
finding a primitive function, or finding an indefinite integral. The second
type are problems which involve adding up a very large number of very
small quantities, (and then taking a limit as the size of the quantities
approaches zero while the number of terms tends to infinity). This process
leads to the definition of the definite integral. Definite integrals are used
for finding area, volume, centre of gravity, moment of inertia, work done
by a force, and in many other applications.
Definition:
The integration denotes the summation of discrete data. The integral is
calculated to find the functions which will describe the area, displacement,
volume, that occurs due to a collection of small data, which cannot be
measured singularly. In a broad sense, in calculus, the idea of limit is used

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where algebra and geometry are implemented. Limits help us in the study
of the result of points on a graph such as how they get closer to each other
until their distance is almost zero.
In Maths, integration is a method of adding or summing up the parts to
find the whole. It is a reverse process of differentiation, where we reduce
the functions into parts. This method is used to find the summation under a
vast scale. Calculation of small addition problems is an easy task which we
can do manually or by using calculators as well. But for big addition
problems, where the limits could reach to even infinity, integration
methods are used. Integration and differentiation both are important parts
of calculus. The concept level of these topics is very high. Hence, it is
introduced to us at higher secondary classes and then in engineering or
higher education. To get an in-depth knowledge of integrals, read the
complete article here.

4. 3
Integration Formulas:
below the formulas of integral or integration, which are commonly used in
higher-level maths calculations. Using these formulas, you can easily solve
any problems related to integration.

5. 4
Theorem:
If F(x) is a particular anti-derivative of f(x) on an interval I, then every
anti-derivative of f(x) on I is given by ∫ f(x) dx = F(x) + C.
 Here, ∫ f(x) dx represents the whole class of integral.
 C is the arbitrary constant, and all the antiderivatives of f(x) on I can
be obtained by assigning a particular value to C.
 Here f(x) is the integrand,
 The variable x in dx is called the integrator and the whole process of
finding the integral is called the integration. The ∫ sign stands for the
sum.
Integration an Inverse Process of Differentiation:
We are given a derivative of a function and are asked to find its primitive,
that is, the original function. Such a process is called anti-differentiation or
integration. If we are given the derivative of a function, the process of
finding the original function is called integration. The derivatives and the
integrals are opposite to each other. Consider a function f(x)= sin x. The
derivative of f(x) is f'(x) = cos x. We say that the function cos x is the
derived function of sin x. Similarly, we say that sin x is the anti-derivative
of cos x.
Methods of Integration:
Sometimes, the inspection is not enough to find the integral of some
functions. There are additional methods to reduce the function in the
standard form to find its integral. Prominent methods are discussed below.
The methods of integration are:
 Decomposition method
 Integration by Substitution

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 Integration using Partial Fractions
Method 1: Integration by Decomposition
The functions can be decomposed into a sum or difference of functions,
whose individual integrals are known. The given integrand will be
algebraic, trigonometric or exponential or a combination of these
functions.
Suppose we need to integrate (x2
-x +1)/x3
dx, we decompose the function
as :
∫ (x2
-x +1)/x3
dx = ∫ (x2
/x3
- x /x3
+1/x3
)
= ∫ (1/x)dx - ∫ (1/x2
) dx + ∫ (1/x3
)dx
Applying the reciprocal rule and the power rule, we get
∫ (x2
-x +1)/x3
dx = log|x| + 1/x - 1/2x2
+ C
Method 2: Integration by Substitution
The integration by substitution method lets us change the variable of
integration so that the integrand is integrated in an easy manner.
Suppose, we have to find y =∫ f(x) dx.
Let x=g(t). Then, dx/dt=g′(t)
So, y= ∫ f(x) dx can be written as y= ∫ f(g(t)) g'(t).
For example, let's find the integral of f(x) = sin(mx) using substitution.
Let mx = t. Then, m(dx/dt)=1
y=∫ sin mxdx
=1m∫ sint dt
=−1/mcost+C
=−1/mcosmx+C
y=∫ sin(mx)dx can be written as ∫ f(g(t)) g'(t)dt

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Note: The substitution for the variable of integration can also use
trignonometric identities. A few important standard results are:
 ∫ tan x dx = log|secx| +C
 ∫ cot x dx = log|sin x| +C
 ∫cosec x dx = log|cosec x -cot x| +C
 ∫ sec x dx = log|secx + tan x| +C
Method 3: Integration using Partial Fractions
By using partial fraction we have 1/ (x+1)(x+2)=A/x+1+B/x+2
We will determine the values of A and B.
we get 1=A / (x+2) + B / (x+1).
From this, we have a set of two linear equations.
A+B=0 and 2A+B =1
On solving these equations we get, A=1 and B=-1.
So, equation (1) can be written as 1/ (x+1)(x+2)=1/(x+1) − 1/(x+2)
Now, solving the integral
∫ (1/ (x+1)(x+2))dx
=∫(1/(x+1) − 1/(x+2))dx
=log|x+1|−log|x+2|+C
=log∣x+1/x+2∣+C
Real-life applications of integration:
Integration has numerous real-life applications across various fields. Here
are some specific examples:
1- Physics and Engineering:
- Motion and Dynamics: Integration is used to calculate displacement,
velocity, and acceleration of objects in motion.

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- Work and Energy: Integration helps determine the work done by a
force and the potential energy stored in a system.
- Fluid Mechanics: Integration is applied to calculate fluid flow rates,
pressure, and the distribution of forces in fluid systems.
2- Economics:
- Total Revenue and Cost: Integration is used to find total revenue and
total cost functions, which are essential in economic analysis.
- Consumer Surplus: Integration is applied to calculate consumer surplus
in economic models.
3- Statistics:
- Probability Distributions: Integration is used to compute probabilities
by finding the area under probability density functions.
- Expected Values: Integration is employed to calculate expected values
in statistics.
4- Computer Graphics:
- Computer-Aided Design (CAD): Integration is used to calculate
volumes and surface areas, which is crucial in designing 3D models in
CAD.
5- Biology:
- Population Growth: Integration is applied to model and analyze
population growth and decay in biological systems.
6- Chemistry:
-
** Reaction Rates: Integration is used to determine reaction rates and
concentrations in chemical kinetics.
7- Medicine:

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- Drug Concentrations: Integration is applied to model the concentration
of drugs in the body over time.
8- Environmental Science:
- Pollutant Dispersion: Integration is used to model the dispersion of
pollutants in air, water, or soil.
9- Telecommunications:
- Signal Processing: Integration is used in signal processing to analyze
and manipulate signals, for instance, in filtering and modulation.
11- Finance:
- Present Value: Integration is applied to calculate the present value of
future cash flows in finance.
- Option Pricing: Integration is used in financial mathematics to derive
option pricing formulas.

10. 9
Find the integral of the function: ∫x2
dx
Solution:
Given ∫x2 dx = (x3
/3) + C.
Integrate ∫ (x2
-1)(4+3x)dx.
Solution:
Given: ∫ (x2
-1)(4+3x)dx.
Multiply the terms, we get
∫ (x2
-1)(4+3x)dx = ∫ 4x2
+3x3
-3x-4 dx
Now, integrate it, we get
∫ (x2
-1)(4+3x)dx = 4(x3
/3) + 3(x4
/4)- 3(x2
/2) – 4x + C
The antiderivative of the given function ∫ (x2
-1)(4+3x)dx is 4(x3
/3) +
3(x4
/4)- 3(x2
/2) – 4x + C.
Integrate the function f(x)=2x sin(x2
+1) with respect to x.
Solution:
Problem:
Problem:
Problem:

11. 11
Calculate ∫ cos2
x dx
Solution:
Summary:
In conclusion, integration is a powerful mathematical concept with broad
applications in diverse fields. Its ability to model and analyze accumulated
quantities, areas, and volumes makes it an indispensable tool in calculus
and real-world problem-solving. From physics and engineering to
economics, biology, and beyond, integration provides a framework for
understanding and quantifying various phenomena.
The Fundamental Theorem of Calculus establishes a fundamental link
between differentiation and integration, providing a powerful tool for
evaluating definite integrals and finding antiderivatives. This connection
further enhances the practicality of integration in solving complex
problems.
In summary, integration is not only a cornerstone of calculus but also a key
element in the mathematical modeling and analysis of natural, economic,
and scientific processes. Its widespread applications highlight its
significance as a fundamental and versatile mathematical tool.
Problem: