## kirbykirby 2 years ago Integral! (related to gamma function maybe?) $\int\limits_{0}^{X}\frac{\lambda^3}{2}t^2e^{-\lambda t}dt • This Question is Closed 1. kirbykirby Here it is: \[\int\limits_{0}^{X}\frac{\lambda^3}{2}t^2e^{-\lambda t}dt$ I tried doing this trying to relate it to a gamma function but myupper bound is not infinity :S: $=\frac{\lambda^3}{2}\int\limits_{0}^{X}t^2e^{-\lambda t}dt$ $Let : u=\lambda t =>t=u/t$$So: du=\lambda dt$$Bounds: u=0=>t=0 ; u=x=>t=x/\lambda$ $\frac{\lambda^2}{2}\int\limits_{0}^{X/\lambda}(\frac{u}{\lambda})^2e^{-u}du$$=\frac{1}{2}\int\limits_{0}^{X/\lambda}u^{3-1}e^{-u}du$

2. Outkast3r09

why do you not just use tabular method

3. kirbykirby

What's that?

4. Outkast3r09

it's a special type of by parts.. you can do by parts also

5. Outkast3r09

|dw:1358916012425:dw|

6. Outkast3r09

|dw:1358916166540:dw|

7. Outkast3r09

$\int t^2e^{-yt}=\frac{-t^2e^{-yt}}{y}-\frac{2te^{-yt}}{y^2}-\frac{2e^{-yt}}{y^3}+c$

8. kirbykirby

Hmm interesting o_o!! As I was trying to do integration by parts, I realized I have to do it twice. Is this method related to doing int. by parts multiple times?

9. Outkast3r09

http://www.wolframalpha.com/input/?i=integral+of+t%5E2e%5E%7B-yt%7D same answer as mine only simplified and yes Tabular method is when you have something of this sort $\int t^nf(t)dt$ you take the derivatives of the t to thepower until you get to zero giving them alternating signs and integrate the f(t) one more time than how many it takes the t power function goes to zero

10. Outkast3r09

look up tabular method for a derivation of this

11. kirbykirby

Oh wow this is quite neat !! :) thank you so much

12. kirbykirby

I can't believe they never taught this technique..