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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. Tags calculus3, evaluate, funky, gaussian, integral, integrals, result Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ricsi046 Math Software 1 September 6th, 2014 11:30 AM mot1975 Calculus 1 April 15th, 2012 07:59 AM layd33foxx Calculus 2 December 12th, 2011 08:49 PM ChloeG Calculus 1 February 16th, 2011 01:15 PM sobadin Calculus 2 November 13th, 2008 10:24 AM

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