Evaluate $\int^{\infty}_0\frac{x^c}{c^x}\,dx$ with $c>1$

Question

For $c>1$, evaluate, $$\int^{\infty}_0\frac{x^c}{c^x}\,dx$$

I observe that, if I replace $c\to e$, then this is firmly the definition of the gamma function. But I don't have $e$ here.

How to proceed further?

Please help.

Answer 1

Rewrite the integrand as

$$\int^{\infty}_0\dfrac{x^c}{c^{x}}dx = \int_0^\infty \left( \frac{x\ln c }{\ln c}\right)^ce^{-x\ln c}\ dx$$

and then substitute $t=x\ln c$ to convert it to the familiar gamma integral.