Question

How do you integrate `( (arctan(x/2) ) ÷ (4+x²) dx`?

Answer 1

`1/4arctan^2(x/2)+C`

Explanation:

`intarctan(x/2)/(4+x^2)dx`

We can use substitution. Let `u=arctan(x/2)`. Taking the derivative, we see that `du=(1/2)/(1+(x/2)^2)dx=(1/2)/((4+x^2)/4)dx=2/(4+x^2)dx`.

Thus:

`=1/2int(2arctan(x/2))/(4+x^2)dx=1/2intudu=1/2(u^2/2)=u^2/4=1/4arctan^2(x/2)+C`