Question

Integrate the function: $\frac{\cos x}{\sqrt{4-{\sin }^{2}x}}$

Answer 1

Let $I=\int \frac{\cos x}{\sqrt{4-{\sin }^{2}x}}$ ...(1)
Substituting $\sin x=t$
Differentiating with respect to$t$
$\Rightarrow \cos x\frac{dx}{dt}=1\Rightarrow \cos xdx=dt$
From equation (1)
$I=\int \frac{1}{\sqrt{4-t^2}}dt$
$\Rightarrow I={\sin }^{-1}\left(\frac{t}{2}\right)+C$
$[\because \int \frac{1}{\sqrt{a^2-x^2}}dx={\sin }^{-1}\frac{x}{a}+C]$
$\therefore I={\sin }^{-1}\left(\frac{\sin x}{2}\right)+C$
Hence,
$\int \frac{\cos x}{\sqrt{4-{\sin }^{2}x}}dx={\sin }^{-1}\left(\frac{\sin x}{2}\right)+C$
Where $C$ is constant of integration.