<s>[INST] Proving $\int_a^b \frac {x dx}{\sqrt{(x^2-a^2)(b^2-x^2)}} = \frac {\pi}{2} $ Can anybody here help me to prove that 
$$
\int_a^b \frac {x\, \mathrm dx}{\sqrt{(x^2-a^2)(b^2-x^2)}} = \frac {\pi}{2} 
$$
Thanks for your help.
 [/INST] After substituting $y=x^2$, this integral may be solved by a type-III Euler substitution, of the form: 
$$\sqrt{\left(y-a^2\right)\left(b^2-y\right)}=\left(y-a^2\right)t.\tag{1}$$
In my opinion, Euler substitutions offer an elegant alternative to trigonometric substitutions, and they are unfortunately under-taught in basic calculus courses. So when applicable, I like to post solutions using the Euler method for the sake of raising awareness, even for problems such as this one that have already been adequately solved by other methods. Using the substitution relation $(1)$ above, we find,
$$\begin{cases}
t=\frac{\sqrt{\left(y-a^2\right)\left(b^2-y\right)}}{y-a^2},\\
y=\frac{a^2t^2+b^2}{t^2+1},\\
\mathrm{d}y=\frac{2t\left(a^2-b^2\right)}{\left(t^2+1\right)^2}\,\mathrm{d}t,\\
\end{cases}$$
and so the integral evaluates to:
$$\begin{align}
I
&=\int_{a}^{b}\frac{x\,\mathrm{d}x}{\sqrt{\left(x^2-a^2\right)\left(b^2-x^2\right)}}\\
&=\frac12\int_{a^2}^{b^2}\frac{\mathrm{d}y}{\sqrt{\left(y-a^2\right)\left(b^2-y\right)}}\\
&=\frac12\int_{\infty}^{0}\frac{t^2+1}{|b^2-a^2||t|}\cdot\frac{2t\left(a^2-b^2\right)}{\left(t^2+1\right)^2}\,\mathrm{d}t\\
&=\frac{b^2-a^2}{|b^2-a^2|}\int_{0}^{\infty}\frac{\mathrm{d}t}{t^2+1}\\
&=\operatorname{sgn}{\left(b^2-a^2\right)}\cdot\frac{\pi}{2}.\\
\end{align}$$
</s>