
13-16 Calculus – Differentiation and Integration
Comparing the coefficients of powers of x on both sides, we get
3 2 4
4 4
1 1
1 2 1
1
= +
− = + +
− =
A A B
B ( ).Step 5
After solving the above system of linear equations with four unknowns, we get
A A B
1 2 1
2 3 1= = − = −, , ( ).Step 6
Substituting the above values in Eq. (1) and integrating on both sides, we get
x x
x x
dx
x
dx
x
dx
x
dx
x
2
2 2
3 4
2
2
2
3
2
1
2 2
− −
+
∫
=
+
∫
−
+
∫
−
∫
= +
( ) ( )
( )
ln(
Step 7
)) ln .+
+
− +
3
2x
x c
Example 13.19 Evaluate
x
x x
dx
+
− −
∫
1
1 2
3
( ) ( )
.
Solution: Observe that the linear factor (x - 1) is repeated thrice in the denominator
1
1
1
1 1
3
1 2
2
3
3
( ) ( ) ( )
( ).
x
A
x
A
x
A
x−
=
−
+
−
+
−
Step 1
Assign the non-repeated ...