If $F(x)=\frac{x^4-3}{x^4+1}$ is a primitive of $f(x)$ find $\int_{0}^{1} xf(x) dx$

Question

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function.

If $F(x)=\frac{x^4-3}{x^4+1}$ is a primitive of $f(x)$ find $\int_{0}^{1} xf(x) dx$

I literally have no idea how to integrate this.

I tried integrating by parts (and finding the derivative of $F(x)$) but I end up getting a even worse integral...

The correct answer apparently is $-3$.

Answer 1

HINT

Integrating by parts we have

$$\int_{0}^{1} xf(x) dx=[xF(x)]_{0}^{1}-\int_{0}^{1} F(x) dx$$