Area between two curves (Demidovich) I'm trying to solve some problems on definite integrals from Demidovich's book and I'm stuck on calculating the area between two curves defined by:
$$y_1 = \frac{a^3}{a^2 + x^2}, y_2 = 0$$
Any hints will be appreciated.

Assume $a>0$. You are asked to evaluate something like
$$
\int_0^M\frac{a^3}{a^2+x^2}dx,
$$ then performing the change of variable $x=a\tan u$ gives
$$
\int_0^M\frac{a^3}{a^2+x^2}dx=\int_0^{\arctan (M/a)}\frac{a^3}{a^2\color{red}{(1+\tan^2u)}}\cdot a\cdot \color{red}{(1+\tan^2u)}\:du=a^2\arctan \left( \frac{M}a\right).
$$ 
