Chapter 08

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January 27, 2005 11:45

L24-CH08

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CHAPTER 8

Principles of Integral Valuation EXERCISE SET 8.1 1. u = 4 − 2x, du = −2dx, − 2. u = 4 + 2x, du = 2dx,

3. u = x , du = 2xdx,

3 2 1 2

2

1 2

1 1 u3 du = − u4 + C = − (4 − 2x)4 + C 8 8

u du = u3/2 + C = (4 + 2x)3/2 + C

sec2 u du =

1 1 tan u + C = tan(x2 ) + C 2 2

2

4. u = x , du = 2xdx,

tan u du = −2 ln | cos u | + C = −2 ln | cos(x2 )| + C

2

1 − 3

5. u = 2 + cos 3x, du = −3 sin 3xdx,

6. u =

2 2 1 x, du = dx, 3 3 6

du 1 1 = − ln |u| + C = − ln(2 + cos 3x) + C u 3 3

du 2 1 1 = tan−1 u + C = tan−1 x + C 1 + u2 6 6 3

sinh u du = cosh u + C = cosh ex + C

7. u = ex , du = ex dx,

8. u = ln x, du =

1 dx, x

sec u tan u du = sec u + C = sec(ln x) + C eu du = eu + C = etan x + C

2

9. u = tan x, du = sec xdx,

du 1 1 = sin−1 u + C = sin−1 (x2 ) + C 2 2 2 1−u 1 1 1 u5 du = − u6 + C = − cos6 5x + C 11. u = cos 5x, du = −5 sin 5xdx, − 5 30 30 1 + 1 + sin2 x 1 + √1 + u 2 du √ 12. u = sin x, du = cos x dx, = − ln + C = − ln +C u sin x u u2 + 1 2

10. u = x , du = 2xdx,

1 2

14. u = tan−1 x, du = √

du = ln u + u2 + 4 + C = ln ex + e2x + 4 + C 4 + u2 −1 dx, eu du = eu + C = etan x + C

13. u = ex , du = ex dx,

1 1 + x2

1 x − 1, du = √ dx, 2 x−1

2

16. u = x2 + 2x, du = (2x + 2)dx,

1 2

15. u =

17. u =

1 x, du = √ dx, 2 x

√ x−1

eu du = 2eu + C = 2e cot u du =

+C

1 1 ln | sin u| + C = ln sin |x2 + 2x| + C 2 2

√ 2 cosh u du = 2 sinh u + C = 2 sinh x + C

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Chapter 08 by Faheem Ajmal - Issuu