This document provides a worksheet with 12 integration problems to practice integration by substitution. The problems involve both indefinite integrals and definite integrals of functions containing variables like x, z, and α. Students are asked to use appropriate substitutions to evaluate the integrals and obtain the most general antiderivative for indefinite integrals or a number for definite integrals. Some problems require determining the substitution, while others provide it. Trigonometric substitutions or identities may also be needed.

1. Worksheet for Section 5.5
Integration by Substitution
V63.0121, Calculus I
December 7, 2009
Find the following integrals. In the case of an indefinite integral, your answer should be the
most general antiderivative. In the case of a definite integral, your answer should be a number.
In these problems, a substitution is given.
1. (3x − 5)17 dx, u = 3x − 5
4
2. x x2 + 9 dx, u = x2 + 9
0
√
e x √
3. √ dx, u = x.
x
cos 3x dx
4. , u = 5 + 2 sin 3x
5 + 2 sin 3x
In these problems, you need to determine the substitution yourself.
5. (4 − 3x)7 dx.
π/3
6. csc2 (5x) dx
π/4
3
−1
7. x2 e3x dx
1

2. 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
2z dz
, both by substituting u = z 2 + 1 and u =
3
9. Find √
3
z 2 + 1.
z2 + 1
Use the trigonometric identity
cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α
to find
10. sin2 x dx
11. cos2 x dx
‘ 12. Find
sec x dx
by multiplying the numerator and denominator by sec x + tan x.
2