Question

Evaluate $\int \frac{dx}{{(a^2+x^2)}^{3/2}}$
$I=\frac{x}{a^2{(x^2+a^2)}^K}+c$
What is K?

Answer 1

$I=\int \frac{dx}{{(a^2+x^2)}^{3/2}}$
Subtitute $x=atan\theta$ or $dx=ase{c}^2\theta d\theta$
$\therefore I=\int \frac{ase{c}^2\theta }{{(a^2+a^{2}ta{n}^2\theta )}^{3/2}}d\theta$
$=\int \frac{ase{c}^2\theta }{a^3{\left(se{c}^2\theta \right)}^{3/2}}d\theta$
$I=\frac{1}{a^2}\int \frac{d\theta }{sec\theta }=\frac{1}{a^2}\int cos\theta d\theta =\frac{1}{a^2}sin\theta +c$
$I=\frac{x}{a^2{(x^2+a^2)}^{1/2}}+c$