These problems demonstrate techniques for integration by substitution. Several integrals are solved by making appropriate substitutions to simplify the integrands, including substituting u = 3x - 5, u = x^2 + 9, u = √x, u = 5 + 2sin3x, and u = 4 - 3x. Trigonometric identities are also used to rewrite sin^2x and cos^2x terms before integrating. One integral is solved using long division and another using two different substitutions. The last integral is solved by multiplying the numerator and denominator by secx + tanx before making the substitution u = tanx + secx.
OBJECTIVES
Revision On:
Simplify of Algebraic Fraction
Perform Operations on Algebraic Fraction
Solve Equations Involving Algebraic Fraction
Make Substitution in Algebraic Fraction
Solve Simultaneous Equations Involving Algebraic Fraction
Undefined value of an Algebraic Fraction
Represent Algebraic Fractions Graphically.
OBJECTIVES
Revision On:
Simplify of Algebraic Fraction
Perform Operations on Algebraic Fraction
Solve Equations Involving Algebraic Fraction
Make Substitution in Algebraic Fraction
Solve Simultaneous Equations Involving Algebraic Fraction
Undefined value of an Algebraic Fraction
Represent Algebraic Fractions Graphically.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
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 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
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.
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.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Lesson 20: Derivatives and the Shapes of Curves (handout)
Lesson 28: Integration by Substitution (worksheet solutions)
1. Solutions to Worksheet for Section 5.5
Integration by Substitution
V63.0121, Calculus I
Summer 2010
1. (3x − 5)17 dx, u = 3x − 5
1
Solution. As suggested, we let u = 3x − 5. Then du = 3 dx so dx = du. Thus
3
1 1 1 18 1
(3x − 5)17 dx = u17 du = · u +C = (3x − 5)18 + C
3 3 18 54
4
2. x x2 + 9 dx, u = x2 + 9
0
Solution. We have du = 2x dx, which takes care of the x and dx that appear in the integrand.
The limits are changed to u(0) = 9 and u(4) = 25. Thus
4 25
1 25 √ 1 2
x x2 + 9 dx = u du = · u3/2
0 2 9 2 3 9
1 98
= · (125 − 27) = .
3 3
√
e x √
3. √ dx, u = x.
x
1
Solution. We have du = √ dx, which means that
2 x
√
e x
√ dx = 2 eu du = 2eu + C
x
1
2. cos 3x dx
4. , u = 5 + 2 sin 3x
5 + 2 sin 3x
Solution. We have du = 6 cos 3x dx, so
cos 3x dx 1 du 1 1
= = ln |u| + C = ln |5 + 2 sin 3x| + C
5 + 2 sin 3x 6 u 6 6
In these problems, you need to determine the substitution yourself.
5. (4 − 3x)7 dx.
1
Solution. Let u = 4 − 3x, so du = −3x dx and dx = − du. Thus
3
1 u8 1
(4 − 3x)7 dx = − u7 du = − + C = − (4 − 3x)8 + C
3 24 24
π/3
6. csc2 (5x) dx
π/4
1
Solution. Let u = 5x, so du = 5 dx and dx = du. So
5
π/3 5π/3
1
csc2 (5x) dx = csc2 u du
π/4 5 5π/4
5π/3
1
=− cot u
5 5π/4
1 5π 5π
= cot − cot
5 4 3
√
1 3
= 1+
5 3
3
−1
7. x2 e3x dx
2
3. 1
Solution. Let u = 3x3 − 1, so du = 9x2 dx and dx = du. So
9
3
−1 1 1
x2 e3x dx = eu du = eu + C
9 9
1 3x3 −1
= e .
9
Sometimes there is more than one way to skin a cat:
x
8. Find dx, both by long division and by substituting u = 1 + x.
1+x
Solution. Long division yields
x 1
=1−
1+x 1+x
So
x dx
dx = x −
1+x 1+x
To find the leftover integral, let u = 1 + x. Then du = dx and so
dx du
= = ln |u| + C
1+x u
Therefore
x
dx = x − ln |x + 1| + C
1+x
Making the substitution immediately gives du = dx and x = u − 1. So
x u−1 1
dx = du = 1− du
1+x u u
= u − ln |u| + C
= x + 1 − ln |x + 1| + C
It may appear that the two solutions are “different.” But the difference is a constant, and we
know that antiderivatives are only unique up to addition of a constant.
2z dz
, both by substituting u = z 2 + 1 and u =
3
9. Find √
3
z 2 + 1.
z2 + 1
3
4. Solution. In the first substitution, du = 2z dz and the integral becomes
du 3 2/3 3
√ = u−1/3 du = u + C = (z 2 + 1)2/3
3
u 2 2
In the second, u3 = z 2 + 1 and 3u2 du = 2z dz. The integral becomes
3u2 3 3 3
du = 3u2 du = u + C = (z 2 + 1)2/3 + C.
u 2 2
The second one is a dirtier substitution, but the integration is cleaner.
Use the trigonometric identity
cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α
to find
10. sin2 x dx
Solution. Using cos 2α = 1 − 2 sin2 α, we get
1 − cos 2α
sin2 α =
2
So
1 1
sin2 x dx = − cos 2x dx
2 2
x 1
= − sin 2x + C.
2 4
11. cos2 x dx
Solution. Using cos 2α = 2 cos2 α − 1, we get
1 + cos 2α
sin2 α =
2
So
1 1
sin2 x dx = + cos 2x dx
2 2
x 1
= + sin 2x + C.
2 4
4
5. 12. Find
sec x dx
by multiplying the numerator and denominator by sec x + tan x.
Solution. We have
sec x(sec x + tan x)
sec x dx = dx
sec x + tan x
sec2 x + sec x tan x
= dx
tan x + sec x
Now notice the numerator is the derivative of the denominator. So the substitution u = tan x+sec x
gives
sec x dx = ln |sec x + tan x| + C.
5