Integration involving $\arcsin$ How to integrate the following function:
$$\int_0^a \arcsin\sqrt{a \over a+x}dx$$
By using the substitution $x = a\tan^2\theta$, I managed to write the integral as:
$$2a\int_0^{\pi \over 4}\theta \frac{\sin\theta}{\cos^3\theta}d\theta$$
How would I proceed? Should I use by parts method?

You are on the right track.
By parts,
$$\int\theta \frac{\sin\theta}{\cos^3\theta}d\theta=\frac\theta{2\cos^2(\theta)}-\int\frac{d\theta}{2\cos^2(\theta)}=\frac\theta{2\cos^2(\theta)}-\frac{\tan(\theta)}2.$$
