Page 122 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 5] INTEGRALS 113

CHANGE OF VARIABLES AND SPECIAL METHODS OF INTEGRATION
ð ð 1 tan 1 t ð 3 dx ð csch 2 p ffiffiffi u
3
2 sin x
5.49. Evaluate: (a) x e 3 cos x dx; ðbÞ dt; ðcÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðdÞ p ffiffiffi du,
0 1 þ t 2 1 4x x 2 u
ð 2 dx
(e) 2 .
2 16 x
2
ffiffiffi
e
Ans. (a) 1 sin x 3 þ c; ðbÞ =32; ðcÞ =3; ðdÞ 2coth p u þ c; 1 ln 3.
3 ðeÞ 4
p ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 dx 3 dx x 1
2
ð p ffiffiffi ð
5.50. Show that (a) ; p þ c.
2 3=2 ¼ 12 ðbÞ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x
2
0 ð3 þ 2x x Þ x x 1
ð
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
5.51. Prove that (a) 2 2 1 2 2 2 2
2 2 u a j
u a du ¼ u u a a ln ju þ
ð
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1 2
2
2
2
2
(b) a u du ¼ u a u þ a sin 1 u=a þ c; a > 0.
2
2
ð xdx p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
5.52. Find p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Ans. x þ 2x þ 5 ln jx þ 1 þ x þ 2x þ 5jþ c.
2
x þ 2x þ 5
5.53. Establish the validity of the method of integration by parts.
ð ð
2
3 2x
1 2x
3
5.54. Evaluate (a) x cos 3xdx; ðbÞ x e dx: Ans. (a) 2=9; ðbÞ e ð4x þ 6x þ 6x þ 3Þþ c
3
0
ð 1 1 1 1
2
5.55. Show that (a) x tan 1 xdx ¼ þ ln 2
0 12 6 6
ffiffiffi ffiffiffi
p p p ffiffiffi
ð
2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 7 3 3 3 5 þ 2 7
2 ln .
ðbÞ x þ x þ 1 dx ¼ 4 þ 4 þ 8 p ffiffiffi
2 2 3 3
5.56. (a)If u ¼ f ðxÞ and v ¼ gðxÞ have continuous nth derivatives, prove that
ð ð
uv ðnÞ dx ¼ uv ðn 1Þ u v þ u v ð 1Þ n u vdx
00 ðn 3Þ
0 ðn 2Þ
ðnÞ
called generalized integration by parts.(b) What simplifications occur if u ðnÞ ¼ 0? Discuss. (c)Use (a)to
ð
4
2
4
evaluate x sin xdx. Ans. (c) 12 þ 48
0
ð 1 xdx 2
5.57. Show that .
2 2 ¼ 8
0 ðx þ 1Þ ðx þ 1Þ
x A B Cx þ D
[Hint: Use partial fractions, i.e., assume and find A; B; C; D.]
2 2 ¼ 2 þ x þ 1 þ x þ 1
2
ðx þ 1Þ ðx þ 1Þ ðx þ 1Þ
ð dx
5.58. Prove that ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi ; > 1.
2
0 cos x 1
NUMERICAL METHODS FOR EVALUATING DEFINITE INTEGRALS
ð 1 dx
5.59. Evaluate approximately, using (a)the trapezoidal rule, (b)Simpson’s rule, taking n ¼ 4.
0 1 þ x
Compare with the exact value, ln 2 ¼ 0:6931.
ð =2
2
2
5.60. Using (a)the trapezoidal rule, (b)Simpson’s rule evaluate sin xdx by obtaining the values of sin x
0
at x ¼ 08; 108; ... ; 908 and compare with the exact value =4.
5.61. Prove the (a) rectangular rule, (b) trapezoidal rule, i.e., (16) and (17)of Page 98.
5.62. Prove Simpson’s rule.