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integration by substitution with an absolute value

Hi Guys,

I need help with a problem I cannot get past. The question below gives me the hint to make the substitution to make the integration easier and I understand that I the area under the curve must be one which will help me solve for c. But I can't see how the original equation simplifies to the integral the question provides. Could someone please show me how the original equation simplifies to the given integral once the substitution has been made. Once I understand that I can solve for c.

Attachment 25129

Thanks!

Nappy

Re: integration by substitution with an absolute value

The integral

Can be rewritten as

If you replace

So both integrals are the same this gives

let

and we have that

This can be evaluated by integration by parts, but it is also the the gamma function evaluated at n+1

So this gives

Re: integration by substitution with an absolute value

How would I go about calculating the first moment of the above equation? I tried both ways by first integrating with the multiplication of x to the integral and the other way by tring to calculate the moment generating function and setting t=0 for the first moment. I keep getting to a point where my answer is (n+1/2)! Is it possible to have a (n+1/2)! factorial. The calculator reads error. I'd really appreciate if someone could show me how to calculate the first moment. I need to find all moments. Thanks for the help!

Re: integration by substitution with an absolute value

Your original function, because of the absolute value, is an even function. Multiplying by [itex]\xi[/itex], makes the integrand odd so the integral is 0.

Re: integration by substitution with an absolute value

Re: integration by substitution with an absolute value