52
1 Mathematical Physics
= −
1
5
ln(2x + 1) +
1
5
ln(x − 2) + C
=
1
5
ln
x − 2
2x + 1
+ C
1.42 r
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
sin θd(sin θ) =
a
2
2
sin
2
θ|
1
0 =
a
2
2
Fig. 1.11 Polar diagram of
the curve r
2 = a
2 sin 2θ
1.43 Since x
2
+ 2 occurs twice as a factor, assume
x
3
+ x
2
+ 2
(x 2 + 2) 2 =
Ax + B
(x 2 + 2) 2 +
C x + D
x 2 + 2
On clearing off the fractions, we get
x
3
+ x
2
+ 2 = Ax + B + (C x + D)(x
2
+ 2)
or x
3
+ x
2
+ 2 = C x
3
+ Dx
2
+ ( A + 2C)x + B + 2D
Equating the coefficients of like powers of x
C = 1, D = 1, A + 2C = 0, B + 2D = 2
This gives A = −2, B = 0, C = 1, D = 1
Hence,
x
3
+ x
2
+ 2
(x 2 + 2) 2 = −
2x
(x 2 + 2) 2 +
x
x 2 + 2
+
1
x 2 + 2
(x
3
+ x
2
+ 2)dx
(x 2 + 2) 2
= −
2xdx
(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
4a
3 dx
x 2 + 4a 2 = lim
b=∞
b
0
4a
3 dx
x 2 + 4a 2 = lim
b=∞
2a
2 tan
−1
x
2a
b
0
= lim b=∞
2a
2 tan
−1
b
2a
= 2a
2
.
π
2
= πa
2