How can I solve $\int x\exp(-Tx^a)\,dx$?

Question

How can I solve $$\int x\exp(-T x^a)\,dx$$

($T$ and $a$ are variables.)

In WolframAlpha, the answer is

$$-x^2(Tx^a)^{-2/a}\,\frac{\Gamma(2/a,Tx^a)}a$$

I don't know why.

Answer 1

You are considering $$I=\int x\,e^{-T \, x^a } dx$$ Let $$T \, x^a=y \implies x=\left(\frac{y}{T}\right)^{\frac{1}{a}}\implies dx=\frac{\left(\frac{y}{T}\right)^{\frac{1}{a}-1}}{a T}\,dy$$ this makes $$I=\frac 1 {a T^{\frac 2a}}\int e^{-y} y^{\frac{2}{a}-1} \,dy$$ and then the incomplete gamma function since $$\int e^{-y} y^{\frac{2}{a}-1} \,dy=-\Gamma \left(\frac{2}{a},y\right)$$ Go back to $x$ for the result.