Question

How do you integrate `int ( 1/((x+1)^2+4)) dx` using partial fractions?

Answer 1

`1/2arctan((x+1)/2)+C`

Explanation:

`I=int1/((x+1)^2+4)dx`

This isn't a job for partial fractions, it's a job for a trig substitution. Let `x+1=2tantheta`. This implies that `dx=2sec^2thetad theta`. Substituting in tells us:

`I=int1/(4tan^2theta+4)(2sec^2thetad theta)=1/2intsec^2theta/(1+tan^2theta)d theta`

Since `1+tan^2theta=sec^2theta`:

`I=1/2intd theta=1/2theta+C`

Undoing the substitution `x+1=2tantheta`:

`I=1/2arctan((x+1)/2)+C`