My Math Forum Using the result of the Gaussian Integral to evaluate other funky integrals

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 December 6th, 2017, 07:29 PM #1 Newbie   Joined: Dec 2017 From: California Posts: 1 Thanks: 0 Using the result of the Gaussian Integral to evaluate other funky integrals I evaluated the Gaussian integral using polar substitution, and got an answer of sqrt pi But my professor also asked us to compute the integral e^(-x^2/2) from negative to positive infinity and the integral of x^2(e^x^2) from 0 to infinity. ... and using our results from the previous step in just a few lines -of work. --How do I do that using my answer for part a?
 December 7th, 2017, 04:25 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 I presume you meant that you found $\displaystyle \int_0^\infty e^{-x^2} dx= \sqrt{\pi}$. You can then use the fact that $\displaystyle e^{-x^2}$ is symmetric about x= 0 to immediately see that $\displaystyle \int_{-\infty}^\infty e^{-x^2}dx= 2\sqrt{\pi}$. Then $\displaystyle \int_{-\infty}^{\infty} e^{-x^2/2} dx$ is easy, there's an obvious substitution. Let $\displaystyle u= \frac{x}{\sqrt{2}}$. $\displaystyle du= \frac{1}{\sqrt{2}} dx$ so $\displaystyle dx= \sqrt{2}du$. As x goes to $\displaystyle \infty$ so does u and as x goes to $\displaystyle -\infty$ so does u. The integral becomes $\displaystyle \sqrt{2}\int_{-\infty}^\infty e^{-u^2}du$. For the second, $\displaystyle \int_0^\infty x^2e^{-x^2} dx$, use "integration by parts". Let $\displaystyle u= x$ and $\displaystyle dv= xe^{-x^2}dx$. Once you have that, use integration by parts again. Last edited by skipjack; December 7th, 2017 at 04:45 AM.

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