Question

${\int }_0^a\frac{x}{\sqrt{a^2+x^2}}dx$

Answer 1

$I={\int }_0^a\frac{x}{\sqrt{a^2+x^2}}dx$
Put $t=a^2+x^2$
$\frac{dt}{dx}=2x$
$\frac{dt}{2}=xdx$
$I=\int \frac{dt}{2\sqrt{t}}$
$I=\frac{1}{2}\frac{t^{-1/2+1}}{-1/2+1}+c$
$I=\frac{1}{2}\frac{\sqrt{t}}{1/2}+c$
$I=\sqrt{t}+c$
$I={\left[\sqrt{x^2+a^2}\right]}_0^a$
$=a\sqrt{2}-a$
$I=a(\sqrt{2}-1)$