2(5)] Write
2
2
1
(
1) (
2)
x
x
x
+x
+
+
+
 =
2
A
B + C
2
(
1)
x
x
+x
+
+
Therefore,
x2 + x + 1 = A (x2 + 1) + (Bx + C) (x + 2)
322
MATHEMATICS
Equating the coefficients of x2, x and of constant term of both sides, we get
A + B =1, 2B + C = 1 and A + 2C = 1 Solving these equations, we get
3
2
1
A
, B
and C
5
5
5
=
=
=
Thus, the integrand is given by
2
2
1
(
1) (
2)
x
x
x
+x
+
+
+
 =
2
2
1
3
5
5
5 (
2)
1
x
x
x
+
+
+
+
 = 
2
3
1
2
1
5 (
2)
5
1
x
x
x
+


+


+
+


Therefore,
2
2
1
(
+1) (
2)
x
x
dx
x
+x
+
+
∫
 =
2
2
3
1
2
1
1
5
2
5
5
1
1
dx
x
dx
dx
x
x
x
+
+
+
+
+
∫
∫
∫
=
2
1
3
1
1
log
2
log
1
tan
C
5
5
5
–
x
x
x
+
+
+
+
+
EXERCISE 7 5
Integrate the rational functions in Exercises 1 to 21