7.1. Integration by Substitution

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7.1 Integration by Substitution 1. Let u = x − 8. Then du = dx. Z

3 dx = (x − 8)2

3 du u2

Z

3u−1 +C −1 3 =− +C x−8

=

2. Let u = 2 + x. Then du = dx. Z √

Z

−1

2 + xdx =

(2 + x) 2 dx Z

1

=

u 2 du 3

=

u2

+C

3 2

2 3 = u 2 +C 3 3 2 = (2 + x) 2 +C 3 3. Let u = 2 + x. Then du = dx. Z

1 √ dx = 2+x

Z

−1

(2 + x) 2 dx Z

−1

=

u 2 du 1

=

u2

+C

1 2 1

= 2u 2 +C √ = 2 2 + x +C 4. x2 1 = x−1− x+1 x+1  Z Z  2 x 1 dx = x−1− dx x+1 x+1 x2 = − x − ln|x + 1|+C 2 220

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Chapter 7. Integration Techniques, Solution Key

5. Let u = e−x + 2. Then du = −e−x dx. e−x 1 dx = − du −x e +2 u = −ln|u|+C

Z

Z

= ln|e−x + 2|+C 6. √ 3 t +5 3 5 = 1 + t t2 t  Z  Z √ 3 5 3 t +5 dt = dt 1 + t t2 t 1

=

3t 2 1 2

+ 5ln|t|+C

√ = 6 t + 5ln|t|+C 7. Let u = 3x − 1. Then du = 3 dx.

Z

2 √ dx = 2 3x − 1

1 −1[ 2 u du 3

Z 1

2 u2 = 1 +C 3 2 4√ 3x − 1 +C = 3 8. Let u = sin x. Then du = cos x dx.

Z

Z

sin x cos x dx =

u du

u2 +C 2 1 = sin2 x +C 2 =

9. Let u = cos x. Then du = − sin x dx.

Z

cos x

Z p 1 − cos2 xdx = − u du

u2 +C 2 1 = − cos2 x +C 2 =−

10. Let u = sin x. Then du = cos xdx. 221

7.1. Integration by Substitution

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Z

sin5 x cos x dx =

Z

u5 du

u6 +C 6 1 = sin6 x +C 6

=

11. Let u = 4x4 . Then du = 16x3 dx. 1 cos u du 16 1 = sin u +C 6
1 = sin 4x4 + c 6

Z

Z

x3 cos(4x4 )dx =

12. Let u = 2x + 4. Then du = 2 dx. 1 sec2 u du 2 1 = tan u +C 2 1 = tan(2x + 4) +C 2

Z

Z

sec2 (2x + 4)dx =

13. Let u = x2 . Then du = 2x dx. Then if x = 0, then u = 0 and if x = 2, then u = 4. Z2

1 xe dx = 2 x2

0

Z4

eu du

0

1 = [eu ]40 2 1 = [e4 − 1] 2 14. Let u = x2 . Then du = 2x dx. √ If x = 0 then u = 0. If x = π, then u = π. √

Zπ

1 x sin x dx = 2

0

2




Zπ

sin u du 0

1 = − [cos u]π0 2 1 = − (cos π − cos 0) 2 1 = − (−1 − 1) 2 =2 222

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Chapter 7. Integration Techniques, Solution Key

15. Let u = x + 5. Then du = dx and x = u − 5. If x = 0, then u = 5. If x = 1 then u = 6.

Z1

Z6

4

x(x + 5) dx = 0

(u − 5)u4 du

5

Z6

=

(u5 − 5u4 ) du

5

u6 u5 = −5 6 5 

6 5

56 = 7776 − 7776 − − 55 6 5 = 520 6 



223