APPM 4360/5360 Homework Question #10 Spring 2015 Problem#1 (55 points): Evaluate the following r

APPM 4360/5360 Homework Assignment #10 Spring 2015 Problem#1 (55 points): Evaluate the following real integrals (where a, b, and k are real and positive): (a) Z 1 0 dx ¡x2 Åa2¢2 (b) Z 1 0 dx (x2 Åa2)(x2 Åb2) (c) Z 1 0 dx x6 Å1 (d) Z 1 ¡1 coskx (x2 Åa2)(x2 Åb2) dx (e) Z 1 ¡1 x coskx x2 Å4x Å5 dx (f ) Z 1 0 coskx x4 Å1 dx (g) Z 1 0 x3 sinkx x4 Åa4 dx (h) Z 2¼ 0 dµ 1Åcos2 µ (i) Z ¼/2 0 sin4 µdµ (j) Z 2¼ 0 dµ (5¡3sinµ)2 (k) Z 1 ¡1 x dx (x2 Åx Å1)2 Problem#2 (10 points): Show that Z 1 0 coshax cosh¼x dx Æ 12 sec³ a2 ´ , jaj Ç ¼.

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APPM4360/5360 HomeworkAssignment#10 Spring2015
Problem#1(55points): Evaluate the following real Use a rectangular contour with corners at§R and
integrals (wherea,b, andk are real and §RÅi .
positive):
Z
1
dx
Problem#3(8points): Use a sector contour with
(a)
¡ ¢
2
2 2
radiusR as in ?gure 4.2.6, centered at the origin with
0
x Åa
2¼
Z angle 0·µ· , to ?nd, foraÈ 0,
1
5
dx
(b)
2 2 2 2
(x Åa )(x Åb )
0 Z
1
dx ¼
Z
1
Æ .
dx
¼
5 5 4
x Åa 5a sin
(c) 0
5
6
x Å 1
0
Z
1
coskx
(d) dx
2 2 2 2
Problem#4(10points): Leta,b2C whereC is the
(x Åa )(x Åb ) r r
¡1
Z
open right half plane. Compute
1
x coskx
(e) dx
2
x Å 4xÅ 5 Z
¡1
1
dx
Z
1
coskx
2 2 2 2
(f) dx ¡1
(x Åa )(x Åb )
4
x Å 1
0
Z
3
1
x sinkx
?rst fora6Æb and then foraÆb.
(g) dx
4 4
x Åa
0
Z
2¼
dµ
Problem#5(17points): Compute the
(h)
2
1Å cos µ
0
following:
Z
¼/2
Z
4
1
(i) sin µdµ dx
(a)
0
2n
x Å 1
¡1
Z
2¼
dµ
Z
(j) 2
1
2 x dx
(5¡ 3sinµ)
0
(b)
2n
Z
x Å 1
¡1
1
xdx
(k)
2 2
(x ÅxÅ 1)
¡1
Extra-CreditProblem#6(8points): Show that
Problem#2(10points): Show that
Z Z
³ ´
1 1
coshax 1 a sinx ¼
dx Æ sec , jajÇ¼. Æ
cosh¼x 2 2 x 2
0 0