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I'm trying to find $$\displaystyle \int{e^{-\frac{y^2}{2}}} \left(\frac{1}{y^2}+1\right)dy$$ I tried using integration by parts and some substitutions, but nothing seem to work.

The answer is $\displaystyle -\frac{{e^{-\frac{y^2}{2}}}}{y}$.

Please help, thank you.

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  • $\begingroup$ Well for a start, you could derive the answer and see that it fits. $\endgroup$ – mvggz Oct 23 '14 at 8:02
  • $\begingroup$ Galc127 I think that there should be a negative sign in the answer. $\endgroup$ – Sherlock Holmes Oct 23 '14 at 8:05
  • $\begingroup$ @SherlockHolmes, absolutely right, thanks for noticing. $\endgroup$ – Galc127 Oct 23 '14 at 8:05
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You must use integration by parts.

First split the integrand into $$\int \frac{e^{\frac{-y^2}{2}}}{y^2}dy + \int e^{\frac{-y^2}{2}}dy$$

Using by parts on the first integral, $$\int \frac{e^{\frac{-y^2}{2}}}{y^2} dy=-\frac{e^{\frac{-y^2}{2}}}{y}-\int e^{\frac{-y^2}{2}}dy$$

This nicely cancels out with the second integral and yields the required result.

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  • $\begingroup$ A great answer, thanks. BTW, you have a little mistake in the last line - it should be a (-) sign instead of (+) sign. $\endgroup$ – Galc127 Oct 23 '14 at 8:10
  • $\begingroup$ @Galc127 Thanks for noticing that but it appears someone has beaten me to editing it. $\endgroup$ – Sherlock Holmes Oct 23 '14 at 8:39

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