Gamma Function basic Integral: $\int_0^{+\infty}f'(t)dt=0$

Question

Let $a>0$

$f(t):= \frac{1}{a+1}e^{-t}t^{a+1}$

show that:

$\int_0^{+\infty}f'(t)dt=0$.

In order to prove the basic well know properties of the Gamma Function my book states that this integral is $0$ not saying how to solve it. Then the properties follow with ease using induction and integration by parts. Note that this is an algebra book, so it takes for granted that I can solve an integral, but this is not the case! I could take this as a "well known fact" too I guess, but I'm curious to see how to solve it!

Thanks!

Answer 1

$$ \int_0^{+\infty}f'(t)\,dt=\lim_{R\to+\infty}\int_0^Rf'(t)\,dt=\lim_{R\to+\infty}(f(R)-f(0))=\lim_{R\to+\infty}f(R)=0. $$