Prove the next integral converges for any $x>0$: $\int_0^{\infty}t^{x-1}e^{-t}dt$ Prove the next integral converges for any $x>0$:
$$\int_0^{\infty}t^{x-1}e^{-t}dt$$
I can't find a proper way to prove that, But what i did so far was:


*

*integration by parts: 
$$\frac{e^{-t}t^x}{x}|_0^{\infty}+ \frac1x\int_0^{\infty}t^{x}e^{-t}dt$$
(My idea behind this part was maybe finding an Indefinite integral i can use)

*$\frac{e^{-t}t^x}{x}|_0^{\infty}=0$ so i'm left only with: $\frac1x\int_0^{\infty}t^{x}e^{-t}dt$

*The interesting thing about the above, is that it's some-kind of recursion where $x\int(x)=\int(x+1)$

*Although it's interesting it doesn't help in my converging proving, Any ideas? 

 A: There is no need to do an integration by parts, though we mention later how it can be useful.
Break up the integral into (say) the integral from $0$ to $1$ and the integral from $1$ to $\infty$.
For the integral from $0$ to $1$, note that the integrand is $\le et^{x-1}$ over the interval. Then use Comparison. There is no trouble at all if $x-1\ge 0$. For $-1\lt x-1\lt 0$, use the known fact that $\int \frac{dt}{t^p}$ converges if $p\lt 1$. This is a basic fact that probably has already been proved in your course. If it hasn't, use the Integral Test.
For the integral from $1$ to $\infty$, rewrite $e^{-t}$ as $e^{-t/2}e^{-t/2}$, and use the fact that $\lim_{t\to\infty}t^{x-1}e^{-t/2}=0$ (you only need the fact the expression is bounded). Then we do a comparison with $\int_1^\infty e^{-t/2}$, which clearly converges.
Integration by parts can be used to bypass the need to separate the integral into two parts, since $t^xe^{-t}$ is "well-behaved" at $0$.
A: Simply we have
$$t^{x-1}e^{-t}\sim_0 t^{x-1}$$
and 
$$\int_0^1t^{x-1}dt \,\,\,\text{is convergent}\iff x>0$$
moreover
$$t^{x-1}e^{-t}=_\infty o\left(\frac{1}{t^2}\right)$$
so
$$\text{the integral}\,\,\,\int_1^\infty t^{x-1}e^{-t}dt \,\,\,\ \text{is convergent}\,\,\forall x\in\mathbb{R}$$
hence we have
$$\text{the integral}\,\,\,\int_0^\infty t^{x-1}e^{-t}dt \,\,\,\ \text{is convergent if}\,\, x>0$$
