Download Integration by Substitution

Integration by Substitution
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 5.3 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource)
and your lecture notes.
EXPECTED SKILLS:
• Know how to simplify a “complicated integral” to a known form by making an appropriate substitution of variables.
PRACTICE PROBLEMS:
For problems 1-21, evaluate the given indefinite integral and verify that your
answer is correct by differentiation.
Z
1.
3x2 (x3 + 3)3 dx
1 3
(x + 3)4 + C
4
Z
5
dx
2.
5x + 3
ln |5x + 3| + C
Z
3.
2x cos (x2 ) dx
sin (x2 ) + C
Z
4.
4x(x2 + 6)2 dx
2 2
(x + 6)3 + C
3
Z
5.
sec (4x) tan (4x) dx
1
sec (4x) + C
4
Z
6. (3x − 5)9 dx
1
(3x − 5)10 + C
30
1
Z
7.
e−2x dx
1
− e−2x + C
2
Z
sin x cos x
8.
dx
1 + sin2 x
1
ln (1 + sin2 x) + C
2
Z
x
x
9.
csc
cot
dx
2
2
x
−2 csc
+C
2
Z
√
10.
−3x3 1 − x4 dx
3
1
(1 − x4 ) 2 + C
2
Z √x
e
1
√ − cos 4x dx
11.
x 4
√
2e
x
−
1
sin (4x) + C
16
Z
1
dx
2 + 4x2
√
1
√ arctan ( 2x) + C
2 2
Z
4x
13.
dx
(3 + x2 )2
12.
−2(3 + x2 )−1 + C
Z
√
14.
x2 4 − x dx.
−
32
16
2
(4 − x)3/2 + (4 − x)5/2 − (4 − x)7/2 + C
3
5
7
2
Z
15.
1
dx (HINT: Complete the square)
+ x − x2
1
+C
arcsin x −
2
q
Z
16.
3
4
e3/x
dx
x2
1
− e3/x + C
3
Z
ex
dx
17.
e2x + 1
tan−1 (ex ) + C
Z
18. (sin 4x)(cos 4x)2/3 dx
−
Z
19.
3
(cos 4x)5/3 + C
20
3
csc2 (3x) tan2 (3x) + x2 ex dx
1 3
1
tan (3x) + ex + C
3
3
Z
1
20.
dx
x ln x
ln | ln x| + C
cos−1 x
√
dx
1 − x2
2
1
− cos−1 x + C
2
Z
21.
22. Use an appropriate
trigonometric identity followed by a reasonable substitution to
Z
evaluate tan x dx
ln | sec x| + C
3
1
32x2 + 77x + 49
2
−
=
. Use this fact to evaluate
23. It can be shown that
2
(3x + 1)(4x + 5)
3x + 1 (4x + 5)2
Z
32x2 + 77x + 49
dx.
(3x + 1)(4x + 5)2
2
1
ln |3x + 1| +
+C
3
4(4x + 5)
π
π
24. Using the substitution x = sin θ for − ≤ θ ≤ , evaluate
2
2
your answer completely in terms of the variable x.
Z √
1 − x2 dx. Express
HINT - The following trigonometric identities will be helpful: sin2 θ + cos2 θ = 1,
1
cos2 θ = (1 + cos (2θ), and sin (2θ) = 2 sin θ cos θ
2
1
1 √
x 1 − x2 + sin−1 x + C
2
2
4