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

1 Answer

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