Integrale formula chiusa

[giammarco.cugliari]

Salve ragazzi, non riesco più a trovare la formula che avevo per calcolare questo tipo di integrali


$ int_(0)^(oo) r^me^(-cr)dr $


so svolgerlo ma ne devo calcolare veramente troppi.

Un grande abbraccio a chiunque mi darà una mano

[javicemarpe]

Try to integrate by parts and use some induction on $m$.

[giammarco.cugliari]

i know how to integrate by parts , i'm looking for faster method

[javicemarpe]

Good luck

[giammarco.cugliari]

[javicemarpe]

I think that integration by parts and induction on $m$ is the fastest method (if it's not the only one).

[giammarco.cugliari]

like this

$ int_(0)^(oo) x^(2k)e^(-cx^2) dx =(1\cdot 3\cdot 5\cdot (2k-1))/(2^(k+1)c^(k+1/2) $

i have to do too many of this with different coefficents

[javicemarpe]

Well, of course the result of integrating by parts will depend on $m$ and $c$ and will give you a closed formula. But first you have to prove it.

[giammarco.cugliari]

i'll try..I thought to find it done

[javicemarpe]

It is not dificult. In fact, it's enough (if I'm not mistaken) to integrate by parts two times: one for the induction and one for the base case.

[giammarco.cugliari]

I found

$ int_(0)^(oo) x^(n)e^(-ax)dx =(n!)/(a^(n+1 $

It was not that much difficult,my laziness..

[javicemarpe]

Congratulations!
You have earned a trophy!

Gamma function

[giammarco.cugliari]

Damn ! My buggy head ..