# Interesting integrals, which I think involve the gamma function

by MeMoses
Tags: function, gamma, integrals, interesting, involve
 P: 129 1. The problem statement, all variables and given/known data Evaluate the intergrals: a) integral of 3^(-4*z^2) dz from 0 to infinity b) integral of dx/(sqrt(-ln(x))) from 0 to 1 c) integral of x^m * e^(-a*x^n) dx from 0 to infinity 2. Relevant equations gamma(n) = integral of e^(-w) * w^(n-1) dw from 0 to infinity 3. The attempt at a solution I'm assuming I use: gamma(n) = integral of e^(-w) * w^(n-1) dw from 0 to infinity as it was used in the rest of the problems in this set, however I have no idea where to begin on (a) and (b). For (c) I've been trying to get it to match up with the gamma function above, but the a variable is giving me some difficulty. I should be able to get (c) eventually, but if you could please help with (a) and (b) that would be great as well with anything that simplifies (c)
 Sci Advisor HW Helper Thanks P: 25,174 For a) change 3^(-4*z^2) to e^(log(3)*(-4)*z^2), now try the substitution w=log(3)*4*z^2. For b) the obvious thing to try is w=(-ln(x)), do you see why?
 P: 129 for a) once w=4z^2*ln(3), dw=8ln(3)z, so then i get 1/(8ln(3)) integral of e^(-w)*z^-1 dw. So do I solve w^(n-1) = z^-1 for n and take the gamma function of n? That just seems wrong/overly comlicated for me. As for b) I'm not sure what there is to see, and when do the limits change to match those on the integral on the gamma function? Also, I'm currently stuck on c) as well
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P: 7,416

## Interesting integrals, which I think involve the gamma function

 Quote by MeMoses 1. The problem statement, all variables and given/known data Evaluate the intergrals: a) integral of 3^(-4*z^2) dz from 0 to infinity b) integral of dx/(sqrt(-ln(x))) from 0 to 1 c) integral of x^m * e^(-a*x^n) dx from 0 to infinity ...however I have no idea where to begin on (a) and (b). For (c) I've been trying to get it to match up with the gamma function above, but the a variable is giving me some difficulty. I should be able to get (c) eventually, but if you could please help with (a) and (b) that would be great as well with anything that simplifies (c)
For (a):

Let $u=2(\sqrt{ln(3)})z\,.$

Look at http://en.wikipedia.org/wiki/Gaussian_integral if you don't know the result for the Gaussian Integral.
Emeritus
HW Helper
PF Gold
P: 7,416
 Quote by MeMoses 1. The problem statement, all variables and given/known data Evaluate the intergrals: a) integral of 3^(-4*z^2) dz from 0 to infinity b) integral of dx/(sqrt(-ln(x))) from 0 to 1 c) integral of x^m * e^(-a*x^n) dx from 0 to infinity 2. Relevant equations gamma(n) = integral of e^(-w) * w^(n-1) dw from 0 to infinity 3. The attempt at a solution I'm assuming I use: gamma(n) = integral of e^(-w) * w^(n-1) dw from 0 to infinity as it was used in the rest of the problems in this set, however I have no idea where to begin on (a) and (b). For (c) I've been trying to get it to match up with the gamma function above, but the a variable is giving me some difficulty. I should be able to get (c) eventually, but if you could please help with (a) and (b) that would be great as well with anything that simplifies (c)
For (b):

Do integration by parts.
Choose u & dv in a somewhat backwards way.

This is what you want for v : $\displaystyle v =\sqrt{-ln(x)}\,.$

Then find dv and find what u must be.

Why do you want $\displaystyle v =\sqrt{-ln(x)}\,?$
Look at the graph of $f(x)=\sqrt{-ln(x)}\,.$

The integral $\displaystyle \int_0^1 f(x)\,dx$ is the same as $\displaystyle \int_0^\infty f^{-1}(y)\,dy$

$\displaystyle f^{-1}(y)=\underline{\ ?\ }$
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P: 25,174
 Quote by SammyS For (a): Let $u=2(\sqrt{ln(3)})z\,.$ Look at http://en.wikipedia.org/wiki/Gaussian_integral if you don't know the result for the Gaussian Integral.
Hey SammyS, you can do all of these by substituting and putting into the form of a gamma function. Your alternative works, but the gamma route doesn't involve any integration by parts.
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