Substitution Method Worksheet With Answers Page 3
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Solutions to Worksheet 5.6
1. Evaluate the following indefinite integrals.
9
a.
x(x + 1)
dx
Let u = x + 1. Then x = u
1 and du = dx. Hence
9
9
10
9
x(x + 1)
dx =
(u
1)u
du =
(u
u
) du
1
1
1
1
11
10
11
10
=
u
u
+ C =
(x + 1)
(x + 1)
+ C.
11
10
11
10
b.
sin(2x
4) dx
1
Let u = 2x
4. Then du = 2 dx or
du = dx. Hence
2
1
1
1
sin(2x
4) dx =
sin u du =
cos u + C =
cos(2x
4) + C
2
2
2
3
x
c.
dx
(x
4
+ 1)
4
1
4
3
3
Let u = x
+ 1. Then du = 4x
dx or
du = x
dx. Hence
4
3
x
1
1
1
1
3
4
3
dx =
du =
u
+ C =
(x
+ 1)
+ C
4
4
4
(x
+ 1)
4
u
12
12
√
d.
4x
1 dx
1
Let u = 4x
1. Then du = 4 dx or
du = dx. Hence
4
√
1
1
2
1
1/2
3/2
3/2
4x
1 dx =
u
du =
·
u
+ C =
(4x
1)
+ C
4
4
3
6
2
e.
x cos(x
) dx
1
2
Let u = x
. Then du = 2x dx or
du = x dx. Hence,
2
1
1
1
2
2
x cos(x
) dx =
cos u du =
sin u + C =
sin(x
) + C
2
2
2
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