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Lavender786 December 6th, 2017 07:29 PM

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?

Country Boy December 7th, 2017 04:25 AM

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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