Question

How do you find the antiderivative of `int x^3/(4+x^2) dx`?

Answer 1

`I=intx^3/(4+x^2)dx`

Let `u=4+x^2`, implying that `du=2xdx`. Also note that `x^2=u-4`. Rearranging the integral:

`I=1/2int(x^2(2xdx))/(4+x^2)=1/2int(u-4)/udu=1/2int(1-4/u)du`

Splitting up the integral:

`I=1/2intdu-2int1/udu=1/2u-2lnabsu=1/2(4+x^2)-2lnabs(4+x^2)+C`

Letting the constant from `1/2(4+x^2)` absorb into `C`:

`I=1/2x^2-2ln(x^2+4)+C`