# How do you integrate int (3 - 8x)/(x(1 - x)) using partial fractions?

Mar 5, 2016

3ln|x| + 5ln|1 - x| + c

#### Explanation:

First write the expression in terms of it's partial fractions.
Since the factors on the denominator are linear then the numerators of the partial fractions will be constants, say A and B.

$\frac{3 - 8 x}{x \left(1 - x\right)} = \frac{A}{x} + \frac{B}{1 - x}$

now multiply both sides by x(1-x)

so 3-8x = A(1-x) + Bx ................................(1)

The aim now is to find the values of A and B . Note that if x = 0 , the term with B will be zero and if x = 1 the term with A will be zero. This is the starting point in finding A and B.

let x = 0 in (1) : 3 = A

let x = 1 in (1) : -5 = B

integral can now be written as :

$\int \frac{3}{x} \mathrm{dx} - \int \frac{5}{1 - x} \mathrm{dx}$

$= 3 \ln | x | + 5 \ln | 1 - x | + c$ ( c is constant of integration )