January 27, 2005 11:45
L24-CH08
Sheet number 1 Page number 337
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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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