Question

What is the antiderivative of `x/(1+x^4)`?

Answer 1

The antiderivative of `x/(1+x^4)` is the integral `int x/(1+x^4) dx`

Let `x^2 = tanu=> u = tan^-1(x^2)`

Hence

`2x dx = sec^2(u) du`

`xdx = (1/2) sec^2(u) du`

now the integral becomes

`I=1/2 int (sec^2(u)) (du) / ( 1 + tan^2(u))`

`I=1/2 int sec^2(u) (du) / sec^2(u)`

`I=1/2 int du`

`I=(1/2) u + c`

substitute back `u = tan^-1(x^2)`

`I=(1/2) tan^-1(x^2) + c`