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
[tex]
\int^{x}_{0}\frac{y^{3/2} d y}{(1+y)^{3/2}}
[/tex]
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:
[tex]
\frac{\sqrt{x}(x+3)}{(1+x)^{1/2}}3 \ln ( \sqrt{1+x}+\sqrt{x} )
[/tex]
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.
An integral to calculate

 Posts: 51
 Joined: August 02 2006
 Affiliation: UC Berkeley
An integral to calculate
You were on the right track. Use the substitution [tex]y=\sinh^2 x[/tex] and it all falls out. The integral becomes
[tex]\int dx [2 \cosh^2 x  4 + (2/\cosh^2 x)]=(1/2)\sinh(2x)+x4x+2\sinh x/\cosh x[/tex].
Since [tex]\sinh(2x)=2\sinh x\,\cosh x[/tex] and [tex]x=\sinh^{1}(\sqrt{y})[/tex] then you get the Mathematica result.
[tex]\int dx [2 \cosh^2 x  4 + (2/\cosh^2 x)]=(1/2)\sinh(2x)+x4x+2\sinh x/\cosh x[/tex].
Since [tex]\sinh(2x)=2\sinh x\,\cosh x[/tex] and [tex]x=\sinh^{1}(\sqrt{y})[/tex] then you get the Mathematica result.
An integral to calculate
Thank you Prof.Linder, I get it.