Solving integral with substitution gives wrong result I want to solve the integral:
$$ \int_{-\infty}^{\infty} \frac{x^2}{(1+|x|^3)^4} \ dx.$$
I thought I could solve that via the substitution $u=1+|x|^3$ since then one has the integral:
$$ \int_{-\infty}^{\infty} \frac{1}{3u^4} \ du.$$
But that seems to be wrong (probably the substitution does not work here since it is not differentiable at 0), and that's why this integral is 0, but it should be $\frac29$. So how do I solve this correctly?

The function is even, so rewrite the integral as
$$\int_{-\infty}^\infty \frac{x^2}{(1+|x|^3)^4}dx = 2\int_0^\infty\frac{x^2}{(1+x^3)^4}dx$$
Now try your substitution.
