Complex Analysis - Maths KU

370 Chapter 6 Series and Residues
EXERCISES 6.6 Answers to selected odd-numbered problems begin on page ANS-20.
6.6.1 Evaluation of Real Trigonometric Integrals
In Problems
�
1–12, evaluate the given trigonometric integral.
2π
� 2π
1
1
1.
dθ 2.
0 1+0.5 sin θ 0 10 − 6 cos θ dθ
� 2π
� 2π
cos θ
1
3.
dθ 4.
3 + sin θ 1+3cos2θ dθ
5.
7.
9.
11.
0
� π
0
� 2π
0
� 2π
0
� 2π
0
1
dθ [Hint: Let t =2π− θ.] 6.
2 − cos θ
sin 2 θ
dθ 8.
5 + 4 cos θ
cos 2θ
dθ 10.
5 − 4 cos θ
cos 2 θ
dθ 12.
2 + sin θ
0
� π
0
� 2π
0
� 2π
0
� 2π
0
1
1 + sin 2 θ dθ
cos 2 θ
3 − sin θ dθ
1
cos θ + 2 sin θ +3 dθ
cos 3θ
5 − 4 cos θ dθ
In Problems 13 and 14, establish the given general result. Use Problem 13 to verify
the answer in Example 1. Use Problem 14 to verify the answer to Problem 7.
� π
dθ
aπ
13.
dθ =
0 (a + cos θ) 2
( √ a2 , a>1
− 1) 3
� 2π
sin
14.
2 θ 2π
dθ =
a + b cos θ b2 � √
a − a2 − b2 � , a>b>0
0
6.6.2Evaluation of Real Improper Integrals
In Problems 15–26, evaluate the Cauchy principal value of the given improper
integral. � ∞
1
15.
−∞ x2 dx
− 2x +2
16.
� ∞
1
−∞ x2 − 6x +25 dx
� ∞
1
17.
−∞ (x2 dx
+4) 2 18.
� ∞
x
−∞
2
(x2 � ∞
1
19.
−∞ (x
dx
+1) 2 2 dx
+1) 3 20.
� ∞
x
−∞ (x2 � ∞
2x
21.
−∞
dx
+4) 3 2 − 1
x4 +5x2 dx
+4
22.
� ∞
1
−∞ (x2 +1) 2 (x2 +9) dx
� ∞
x
23.
0
2 +1
x4 dx
+1
24.
� ∞
1
0 x6 +1 dx
� ∞
x
25.
2
x6 dx
+1
26.
� ∞
x 2
(x2 +2x + 2)(x2 dx
+1) 2
0
In Problems 27–38, evaluate the Cauchy principal value of the given improper
integral. � ∞
cos x
27.
−∞ x2 dx
+1
28.
� ∞
cos 2x
−∞ x2 +1 dx
� ∞
x sin x
29.
x2 dx
+1
30.
� ∞
cos x
(x2 dx
+4) 2
−∞
−∞
0