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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: 3,261 Thanks: 894 
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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calculus3, evaluate, funky, gaussian, integral, integrals, result 
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