1. Integration question

Can someone show me how

$\displaystyle \int_{0}^{\infty}t^{k}e^{-at}dt=\frac{k!}{a^{k+1}}\text{ where }k\geq 0 \text{ is an integer and }a>0$

I understand that you repeatedly integrate by parts, but how does everything reduce to $\displaystyle \frac{k!}{a^{k+1}}$?

Thanks

2. Re: Integration question

Originally Posted by downthesun01
Can someone show me how

$\displaystyle \int_{0}^{\infty}t^{k}e^{-at}dt=\frac{k!}{a^{k+1}}\text{ where }k\geq 0 \text{ is an integer and }a>0$

I understand that you repeatedly integrate by parts, but how does everything reduce to $\displaystyle \frac{k!}{a^{k+1}}$?

Thanks
Did you repeatedly integrate it by parts and see what happend ? Show us what you have done.

3. Re: Integration question

Originally Posted by downthesun01
Can someone show me how

$\displaystyle \int_{0}^{\infty}t^{k}e^{-at}dt=\frac{k!}{a^{k+1}}\text{ where }k\geq 0 \text{ is an integer and }a>0$

I understand that you repeatedly integrate by parts, but how does everything reduce to $\displaystyle \frac{k!}{a^{k+1}}$?

Thanks
\displaystyle \displaystyle \begin{align*} \int_0^{\infty}{ t^k \, e^{ -a \, t } \, dt } &= \frac{1}{ a^{k+1} } \int_0^{\infty}{ a^{k+1} \, t^k \, e^{-a\,t} \, dt } \\ &= \frac{1}{a^{k+1}} \int_0^{\infty}{ a \left( a\,t \right) ^k\, e^{-a\,t}\,dt} \\ &= \frac{1}{a^{k+1}}\int_0^{\infty}{ u^k\, e^{-u}\,du } \textrm{ after making the substitution } u = a\,t \implies du = a\,dt \\ &= \frac{1}{a^{k+1}} \int_0^{\infty}{ u^{(k+1) - 1} \, e^{-u}\,du } \\ &= \frac{1}{a^{k+1}}\,\Gamma{ \left( k + 1 \right) } \\ &= \frac{k!}{a^{k+1}} \textrm{ since } \Gamma{ (n)} = (n - 1)! \textrm{ if } n \in \mathbf{Z} \end{align*}

4. Re: Integration question

Here's what I have after integrating by parts 10 times:

$\displaystyle \frac{-t^{k}}{a}e^{-at}+\frac{kt^{k-1}}{a^2}e^{-at}-\frac{k(k-1)t^{k-2}}{a^3}e^{-at}-...+\frac{k(k-1)(k-2)(k-3)(k-4)(k-5)(k-6)(k-7)(k-8)t^{k-9}}{a^{10}}e^{-at}$

5. Re: Integration question

Thank you. Where does the gamma come from?

Nevermind, I looked up the gamma function.

6. Re: Integration question

It's the Gamma Function, which is the continuous function you get if you interpolate the Factorial function so it is defined over most numbers instead of just the positive integers.

7. Re: Integration question

Originally Posted by downthesun01
Here's what I have after integrating by parts 10 times:

$\displaystyle \frac{-t^{k}}{a}e^{-at}+\frac{kt^{k-1}}{a^2}e^{-at}-\frac{k(k-1)t^{k-2}}{a^3}e^{-at}-...+\frac{k(k-1)(k-2)(k-3)(k-4)(k-5)(k-6)(k-7)(k-8)t^{k-9}}{a^{10}}e^{-at}$
You are on the good track !
But in the result of each intagration by part , there is a remaining integral (to be used at the next step). Where is the last remaining integral ?
You did it 10 times. OK. But what happend before, if k=4 for example ?
And do not forget that the limits of the integral are t=0 and t=infinity. Where are they applied into your result ?