How to calculate $\int_{0}^{\infty}\frac{dx}{\sqrt{x^{7}}e^{x}}$?

Question

I am trying to calculate the integral $\int_{0}^{\infty}\frac{dx}{\sqrt{x^{7}}e^{x}}$.

I worked out the answer to be $\Gamma[-\frac{5}{2}]$, However when I used mathematica to check my answer it says the the integral does not converge for $(0,\infty)$ so does that mean that my answer is wrong?

Answer 1

Mathematica is right. This integral does not converge.

The integral representation for $\Gamma$ function $$ \Gamma(z) = \int_0^\infty x^{z-1} e^{-x}\,dx $$ is only valid for $\Re z > 0$.

To show that your integral diverges, observe that $$ \int_\delta^1 x^{-7/2} e^{-x}\,dx \geq \frac{1-\delta}{e} \int_\delta^1 x^{-7/2}\,dx $$ Letting $\delta \to 0$, you can see that your integral does indeed diverge.