Question

Prove $\int \frac{dx}{\sqrt{a^2-x^2}}$

Answer 1

$\int \frac{dx}{a\sqrt{1-{\left(\frac{x}{a}\right)}^2}}=\frac{1}{a}\int \frac{dx}{\sqrt{1-{\left(\frac{x}{a}\right)}^2}}$

Let $\frac{x}{a}=\sin t$

$\Rightarrow dx=a.\cos t.dt$

$=\frac{1}{a}\int \frac{a.\cos t.dt}{\sqrt{1-{\sin }^{2}t}}=\frac{1}{a}\int \frac{a.\cos t.dt}{\cos t}$

$=\int dt=t+c={\sin }^{-1}\left[\frac{x}{a}\right]+c.$where $c$ is the constant of integration.