Prove Reduction Formula

[flaren5]

I don't know how to input the work that I have done on the following question, and I know that we are to show our work. I was able to post the question as an attachment. If any one can give me insight on how to prove the reduction formula with the following question, that would be great, this way I can see if I'm on the right track.

 

[soroban]

Hello, flaren5!

\(\displaystyle \displaystyle\text{Prove: }\;\int x^ne^{\text{-}x^2}dx \;=\;\text{-}\tfrac{1}{2}x^{n-1}e^{-x^2} + \tfrac{n-1}{2}\int x^{n-2}e^{-x^2}dx\)

\(\displaystyle \displaystyle\text{We have: }\;I \;=\;\int x^ne^{\text{-}x^2}dx\)

\(\displaystyle \text{By parts: }\;\begin{Bmatrix}u &=& x^{n-1} && dv &=& xe^{\text{-}x^2}dx \\ du &=& (n-1)x^{n-2}dx && v &=& \text{-}\frac{1}{2}e^{\text{-}x^2} \end{Bmatrix}\)

\(\displaystyle \displaystyle \text{Then: }\;I \;=\; \overbrace{\left(x^{n-1}\right)}^u \overbrace{\left(\text{-}\tfrac{1}{2}e^{\text{-}x^2}\right)}^v - \int\overbrace{\left(\text{-}\tfrac{1}{2}e^{\text{-}x^2}\right)}^v\overbrace{(n-1)x^{n-2}dx}^{du}\)

. . . . . \(\displaystyle \displaystyle I \;=\;\text{-}\tfrac{1}{2}x^{n-1}e^{\text{-}x^2} + \tfrac{n-1}{2}\int x^{n-2}e^{\text{-}x^2}dx\)

 

[flaren5]

Thank you very much for your post, it was very clear and easy to follow. I was glad to see that I was on the right track, but I kept getting stuck on the part that I put a red box around (the derivative of the exponent)...I'm still having a hard time understanding it, such as, where the -1/2 comes from. I know that the deriv of x is 1, but then I'm confused from there. A simple explanation would be greatly appreciated...I think I may be forgetting a rule.