[SOLVED] Amath 567, homework 3

1. Problem 1. From AF: 4.2.1 c d
(a) Evaluate
I =
Z ∞
0
dx
(x
2 + a
2)(x
2 + b
2)
where a
2
, b2 > 0.(b) Evaluate
I =
Z ∞
0
dx
x
6 + 12. Problem 2. From AF: 4.2.2 a, b h. Evaluate the following integrals:
(a) R ∞
−∞
x sin(x)
x2+a2 dx; a
2 > 0(b) R ∞
−∞
cos(kx)dx
(x2+a2)(x2+b
2)
; a
2
, b2
, k > 0(c) R 2π
0
dθ
(5−3 sin θ)
23. Problem 3. From AF: 4.2.7. Use a sector contour with radius R, as
in Figure 4.2.6, centered at the origin with angle 0 ≤ θ ≤
2π
5
to find, fora > 0,
I =
Z ∞
0
dx
x
5 + a
5
=
π
5a
4 sin(π/5)