CosmoCoffee

YinZhe Ma · Posted: December 23 2015

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.

Eric Linder · Joined: 02 Aug 2006 Posts: 48 Affiliation: UC Berkeley

You were on the right track. Use the substitution

y = sinh 2 x

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.

YinZhe Ma · Posted: December 27 2015

Thank you Prof.Linder, I get it.