Question

How do you integrate `int (x+4)/(x^2 + 2x + 5)dx` using partial fractions?

Answer 1

`1/2[ln|x^2+2x+5|+3arctan((x+1)/2)]+c`

Explanation:

First split the integral into two parts:

`x+1` is a scalable factor of the derivative of `x^2+2x+5`, so divide `x+4` by `x+1`

`int(x+4)/(x^2+2x+5)dx=int(x+1)/(x^2+2x+5)dx+int3/(x^2+2x+5)dx`

Algebraic manipulation yields:
`1/2int(2x+2)/(x^2+2x+5)dx+3int1/((x+1)^2+4)dx`
`=1/2ln|x^2+2x+5|+3/2arctan((x+1)/2)+c`
`=1/2[ln|x^2+2x+5|+3arctan((x+1)/2)]+c`