Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

(c) [(x−a)/(b−x)]^1 /^2.

2.38 Determine whether the following integrals exist and, where they do, evaluate
them:
(a)

∫∞

0

exp(−λx)dx;(b)

∫∞

−∞

x

(x^2 +a^2 )^2

dx;

(c)

∫∞

1

1

x+1

dx;(d)

∫ 1

0

1

x^2

dx;

(e)

∫π/ 2

0

cotθdθ;(f)

∫ 1

0

x

(1−x^2 )^1 /^2

dx.

2.39 Use integration by parts to evaluate the following:

(a)

∫y

0

x^2 sinxdx;(b)

∫y

1

xlnxdx;

(c)

∫y

0

sin−^1 xdx;(d)

∫y

1

ln(a^2 +x^2 )/x^2 dx.

2.40 Show, using the following methods, that the indefinite integral ofx^3 /(x+1)^1 /^2 is

J= 352 (5x^3 − 6 x^2 +8x−16)(x+1)^1 /^2 +c.

(a) Repeated integration by parts.

(b) Settingx+1=u^2 and determiningdJ/duas (dJ/dx)(dx/du).

2.41 The gamma function Γ(n) is defined for alln>−1by

Γ(n+1)=

∫∞

0

xne−xdx.

Find a recurrence relation connecting Γ(n+1) and Γ(n).

(a) Deduce (i) the value of Γ(n+1) whennis a non-negative integer, and (ii)

the value of Γ

( 7

2

)

,giventhatΓ

( 1

2

)

=

√

π.

(b) Now, taking factorial( mforanymto be defined bym!=Γ(m+ 1), evaluate

−^32

)

!.

2.42 DefineJ(m, n), for non-negative integersmandn, by the integral

J(m, n)=

∫π/ 2

0

cosmθsinnθdθ.

(a) EvaluateJ(0,0),J(0,1),J(1,0),J(1,1),J(m,1),J(1,n).

(b) Using integration by parts, prove that, formandnboth>1,

J(m, n)=

m− 1

m+n

J(m− 2 ,n)andJ(m, n)=

n− 1

m+n

J(m, n−2).

(c) Evaluate (i)J(5,3), (ii)J(6,5) and (iii)J(4,8).

2.43 By integrating by parts twice, prove thatInas defined in the first equality below
for positive integersnhas the value given in the second equality:

In=

∫π/ 2

0

sinnθcosθdθ=

n−sin(nπ/2)

n^2 − 1

.

2.44 Evaluate the following definite integrals:

(a)

∫∞

0 xe

−xdx;(b)∫^1

0

[

(x^3 +1)/(x^4 +4x+1)

]

dx;

(c)

∫π/ 2

0 [a+(a−1) cosθ]

− (^1) dθwitha> 1
2 ;(d)

∫∞

−∞(x

(^2) +6x+18)− (^1) dx.