Question

Solve $\int \frac{dx}{(x^2+a^2{)}^{3/2}}$

Answer 1

$\int \frac{dx}{(x^2+a^2{)}^{3/2}}$
Let x = a tan y.
$dx=ase{c}^{2}ydy$
$\int \frac{ase{c}^{2}ydy}{(a^{2}ta{n}^{2}y+a^2{)}^{3/2}}$
$=\int \frac{ase{c}^{2}y}{(a^{2}se{c}^{2}y{)}^{3/2}}$
$=\int \frac{ase{c}^{2}y}{a^{3}se{c}^{3}y}$
$=\frac{1}{a^2}\int cosy$
$\frac{1}{a^2}siny+c$
$\frac{x}{a}=tany$
$coty=\frac{a}{x}$
$cosecy=\sqrt{1+\frac{a^2}{x^2}}$
$=\sqrt{\frac{x^2=a^2}{x}}=\frac{1}{a^2}\frac{x}{\sqrt{x^2+a^2}}+c$
$siny=\frac{x}{\sqrt{x^2+a^2}}$