Integration using trig subs...having trouble switching back to the original variable

[Odyssey]

Hi, I carried out the integration until the very end...I don't know how to convert the variable back to the original one. :confused:

\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{a^2-u^2}}

Let u = a\sin{\Theta}
du = a\cos{\Theta}d\Theta

The integral becomes...

\int_{R_{0}}^{R(\Theta)}\frac{a\cos{\Theta}d\Theta }{a\cos{\Theta}a\sin{\Theta}}
\frac{1}{a}\int_{R_{0}}^{R(\Theta)}\csc{\Theta}d{\ Theta}

\csc{\Theta}d{\Theta} = \ln {|\csc{\Theta}-\cot{\Theta}|}+C

This is where I'm stuck. I don't know how to convert the thetas back into the "u"s. I haven't multiplied the answer by 1/a yet. I know that \Theta=\sin^{-1}{u/a}, but if I plug the \sin^{-1}{u/a} into Theta, the expression becomes super messy and I really don't know what to do with it.

Please help, thanks in advance! :smile:

[vsage]

make a triangle.. it's the only way. For example, if theta = arcsin(u/a) then sin(arcsin(u/a)) = u/a, cos(arcsin(u/a)) = sqrt(a^2-u^2)/a. Make a triangle with sides u, sqrt(a^2-u^2) and a. You can find all the trig functions from it (be sure to label theta)