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An integral to calculate
 
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YinZhe Ma



Joined: 09 Oct 2008
Posts: 9
Affiliation: University of British Columbia

PostPosted: December 23 2015  Reply with quote

I have a simple and native equation, but really struggle to find the way to it. Can anyone tell me how to calculate this integral
\int^{x}_{0}\frac{y^{3/2} d y}{(1+y)^{3/2}}
I have trying various different ways to solve it, including changing variables from x to sinh function, cosh function etc., but failed. However, I tried it on mathematica and it indeed can find a solution:
\frac{\sqrt{x}(x+3)}{(1+x)^{1/2}}-3 \ln ( \sqrt{1+x}+\sqrt{x} )

Please let me know if you have a smart way of doing the integral. this integral is crucial to solve the growth factor in the open universe model.
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Eric Linder



Joined: 02 Aug 2006
Posts: 47
Affiliation: UC Berkeley

PostPosted: December 26 2015  Reply with quote

You were on the right track. Use the substitution y = sinh2x and it all falls out. The integral becomes
\int dx [2 \cosh^2 x - 4 + (2/\cosh^2 x)]=(1/2)\sinh(2x)+x-4x+2\sinh x/\cosh x.
Since \sinh(2x)=2\sinh x\,\cosh x and x=\sinh^{-1}(\sqrt{y}) then you get the Mathematica result.
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YinZhe Ma



Joined: 09 Oct 2008
Posts: 9
Affiliation: University of British Columbia

PostPosted: December 27 2015  Reply with quote

Thank you Prof.Linder, I get it.
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