# how to take integral of the Gamma Function

Here is my question. I am trying to calculate the following integral:

$$\int_{t}^{\infty} \Gamma(ax+b)dx$$

I was not successful, although I tried many times.

I wonder if anyone can help me on this?

Thank you.

• The integral diverges. – GEdgar Dec 30 '19 at 11:38

An indefinite integral of $$\Gamma(s) := \int_0^\infty x^{s-1}e^{-x}\;dx,\quad \mathrm{Re}\;s > 0 ,$$ is $$F(s) := \int_0^\infty \frac{(x^{s-1}-1)e^{-x}}{\log x}\;dx,\quad \mathrm{Re}\;s > 0.$$ But $$\lim_{s \to +\infty} F(s) = +\infty$$, so $$\int_t^{+\infty} \Gamma(ax+b)\;dx= +\infty$$ if $$a>1$$.
Also $$\lim_{s \to 0^+} F(s) = +\infty$$, so $$\int_t^{+\infty} \Gamma(ax+b)\;dx= +\infty$$ if $$0 < a < 1$$.