Question

How do we integrate this ?

Answer 1

we are given

$\int \:\frac{4}{\left(x+1\right)\left(x-1\right)^2}dx$

we can use partial fraction method

firstly , we will find partial fraction

$\frac{1}{\left(x+1\right)\left(x-1\right)^2}:\quad \frac{1}{4\left(x+1\right)}-\frac{1}{4\left(x-1\right)}+\frac{1}{2\left(x-1\right)^2}$

$=4\cdot \int \:\frac{1}{4\left(x+1\right)}-\frac{1}{4\left(x-1\right)}+\frac{1}{2\left(x-1\right)^2}dx$

now, we can distribute integral

$=4\left(\int \frac{1}{4\left(x+1\right)}dx-\int \frac{1}{4\left(x-1\right)}dx+\int \frac{1}{2\left(x-1\right)^2}dx\right)$

now, we can solve each

$=4\left(\frac{1}{4}\ln \left|x+1\right|-\frac{1}{4}\ln \left|x-1\right|-\frac{1}{2\left(x-1\right)}\right)$

$=\ln \left|x+1\right|-\ln \left|x-1\right|-\frac{2}{x-1}+C$...........Answer