1000 Solved Problems in Modern Physics

52 1 Mathematical Physics

=−

1

5

ln(2x+1)+

1

5

ln(x−2)+C

=

1

5

ln

(

x− 2

2 x+ 1

)

+C

1.42r^2 =a^2 sin 2θ
Elementary area

dA=

1

2

r^2 dθ

A=

1

2

∫π/ 2

0

r^2 dθ=

a^2

2

∫π/ 2

0

sin 2θdθ=

a^2

∫π/ 2

0

sinθd(sinθ)=

a^2

2

sin^2 θ|^10 =

a^2

2

Fig. 1.11Polar diagram of
the curver^2 =a^2 sin 2θ

1.43 Sincex^2 +2 occurs twice as a factor, assume

x^3 +x^2 + 2

(x^2 +2)^2

=

Ax+B

(x^2 +2)^2

+

Cx+D

x^2 + 2

On clearing off the fractions, we get

x^3 +x^2 + 2 =Ax+B+(Cx+D)(x^2 +2)

orx^3 +x^2 + 2 =Cx^3 +Dx^2 +(A+ 2 C)x+B+ 2 D

Equating the coefficients of like powers ofx

C= 1 ,D= 1 ,A+ 2 C= 0 ,B+ 2 D= 2

This givesA=− 2 ,B= 0 ,C= 1 ,D= 1

Hence,

x^3 +x^2 + 2

(x^2 +2)^2

=−

2 x

(x^2 +2)^2

+

x

x^2 + 2

+

1

x^2 + 2

∫

(x^3 +x^2 +2)dx

(x^2 +2)^2

=−

∫

2 xdx

(x^2 +2)^2

+

∫

xdx

x^2 + 2

+

∫

dx

x^2 + 2

=

1

x^2 + 2

+

1

2

ln(x^2 +2)+

1

√

2

tan−^1

(

x

√

2

)

+C

1.44

∫∞

0

4 a^3 dx

x^2 + 4 a^2

=lim

b=∞

∫b

0

4 a^3 dx

x^2 + 4 a^2

=lim

b=∞

[

2 a^2 tan−^1

(x

2 a

)]b

0

=limb=∞

[

2 a^2 tan−^1

(

b

2 a

)]

= 2 a^2.

π

2

=πa^2