I'm wondering how come the following equality is right? I know the property of the integral $\int_a^b=-\int_b^a$, but the below one is new for me.
$$ \int_{- \infty}^{0} { x^n \, e^x dx} = \int_{0}^{+ \infty} { x^n \, e^{-x} dx} $$
EDIT: $n$ is even.
A sufficient, but not necessary, condition for the following equation to be true $\int_{- \infty}^{0} {g(x) dx} = \int_{0}^{+ \infty} {f(x) dx}$ is that $f(x)=g(-x) \quad if \; x>0 \quad$. Therefore, we could try to prove that $if \; x >0 \quad x^ne^{-x} = ({-x})^ne^{-x}$. After applying some reductions we get $$if \; x >0 \quad x^ne^{-x} = (-1)^{n}({x}^ne^{-x})$$ which is true for all n even. If you want it to be true for all n, you have to multiply for $(-1)^n$ one member of the equation.