Question

How do we integrate this ?

Answer 1

we are given

`\int \:(4)/(\left(x+1\right)\left(x-1\right)^2)dx`

we can use partial fraction method

firstly , we will find partial fraction

`(1)/(\left(x+1\right)\left(x-1\right)^2):\quad (1)/(4\left(x+1\right))-(1)/(4\left(x-1\right))+(1)/(2\left(x-1\right)^2)`

`=4\cdot \int \:(1)/(4\left(x+1\right))-(1)/(4\left(x-1\right))+(1)/(2\left(x-1\right)^2)dx`

now, we can distribute integral

`=4\left(\int (1)/(4\left(x+1\right))dx-\int (1)/(4\left(x-1\right))dx+\int (1)/(2\left(x-1\right)^2)dx\right)`

now, we can solve each

`=4\left((1)/(4)\ln \left|x+1\right|-(1)/(4)\ln \left|x-1\right|-(1)/(2\left(x-1\right))\right)`

`=\ln \left|x+1\right|-\ln \left|x-1\right|-(2)/(x-1)+C`...........Answer