Question

How do you integrate `(x+2)/(x^2+8x+16)` using partial fractions?

Answer 1

`int (x+2)/(x^2+8x+16) dx= ln|x+4|+2/(x+4)+C`

Explanation:

First you factorize the denominator of the rational function:

`x^2+8x+16 = (x+4)^2`

Now you can write:

`(x+2)/(x^2+8x+16) = (x+2)/((x+4)^2)= (x+4-2)/((x+4)^2)=1/(x+4)-2/((x+4)^2)`

The integral is then:

`int (x+2)/(x^2+8x+16) dx= int (dx)/(x+4)-2int(dx)/((x+4)^2)=ln|x+4|+2/(x+4)+C`