Page 765 - Advanced_Engineering_Mathematics o'neil

22.3 Evaluation of Real Integrals 745

EXAMPLE 22.20
We will evaluate
2π
1
dθ
α + β cos(θ)
0
in which 0 <β <α.
Replace cos(θ) = (z + 1/z)/2, and use equation (22.5) to produce the function
1 1 −2i
f (z) = = .
2
α + (β/2)(z + 1/z) iz βz + 2αz + β
f has simple poles at
√
2
−α ± α − β 2
z = .
β
Since α> β, these numbers are real. Only one of them,
√
2
−α + α − β 2
z 1 = ,
β
is enclosed by γ . Therefore,
1
2π
dθ = 2πiRes( f, z 1 )
α + β cos(θ)
0
−2i 2π
= 2πi = √ .
2βz 1 + 2α α − β 2
2
SECTION 22.3 PROBLEMS

In each of Problems 1 through 10, evaluate the integral. 9. ∞ x 2 dx
2
Wherever they appear, α and β are positive numbers. −∞ (x + 4) 2
cos(βx)
∞
10. dx
1 −∞ 2 2 2
2π (x + α )
1. dθ
0 2 − cos(θ)
In Problems 11 through 18, α and β are positive numbers
1
∞ wherever they occur.
2. dx
x + 1
−∞ 4
∞ cos(αx)

−α
1 11. Show that dx = πe .
3. ∞ dx −∞ x + 1
2

6
−∞ x + 1
2
∞ x cos(αx) π
−αβ
2π 1 12. Show that −∞ 2 2 2 dx = e (1 − αβ).
4. dθ (x + β ) 2β
0
6 + sin(θ)
2π 1
13. Let α =β. Show that dθ =
∞ x sin(2x) 0 2
2
2
2
α cos (θ) + β sin (θ)
5. dx
4
−∞ x + 16 2π
.
1 αβ
∞
6. dx
−∞ x − 2x + 6 π/2 1 π
2
14. Show that 0 2 dθ = √ .
2
∞ cos (x) α + sin (θ) 2 α(1 + α)

7. dx √
−∞ 2 2 π
(x + 4) ∞ −x 2 −β 2
15. Show that e cos(2βx)dx = e .
0 2
2sin(θ)
2π −z 2
8. dθ Hint:Integrate e about the rectangular path having
0 2
2 + sin (θ) corners at ±R and ±R + βi. Use Cauchy’s theorem
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October 14, 2010 15:37 THM/NEIL Page-745 27410_22_ch22_p729-750