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- - **Using the result of the Gaussian Integral to evaluate other funky integrals**
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Using the result of the Gaussian Integral to evaluate other funky integralsI 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? |

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. |

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