
13-6 Calculus – Differentiation and Integration
Here, a = 1, b = -5, c = 2 and d = 1
Thus,
1
5 2 1
1
5 2 1 1
5
2 1
1
11
( )( ) ( )( ) ( )( )
ln
l
x x
dx
x
x
c
+ −
∫
=
− −
+
−
+
= −
nn
ln .
x
x
c
x
x
c
+
−
+
=
−
+
+
5
2 1
1
11
2 1
5
Example 13.8 Evaluate
px q
ax b cx d
dx
+
− −
∫
( )( )
.
Solution: It is a proper algebraic rational function with 1 as the degree of numerator and 2 as the
degree of denominator. The denominator has two linear factors (ax - b) and (cx - d).
Assign (ax - b) and (cx - d) to partial fractions
A
ax b
B
cx d- -
, (Step 1).
Considering the equation
px q
ax b cx d
A
ax b
B
cx d
+
− −
=
−
+
−( )( )
(Step 2)
and putting all the numerators on the RHS of the equation ...