Question

Evaluate: $\int \frac{\sin x}{\sqrt{{\cos }^{2}x-2\cos x-3}}dx$

Answer 1

Consider the given integral.
$I=\int \frac{\sin x}{\sqrt{{\cos }^{2}x-2\cos x-3}}dx$
$I=\int \frac{\sin x}{\sqrt{{\cos }^{2}x-2\cos x+1-4}}dx$
$I=\int \frac{\sin x}{\sqrt{(\cos x-1{)}^2-4}}dx$
Let $t=\cos x-1$
$-dt=\sin xdx$
Therefore,
$I=-\int \frac{dt}{\sqrt{t^2-2^2}}$
$I=-\log (t+\sqrt{t^2-4})+C$
On putting the value of $t$, we get
$I=-\log (\cos x-1+\sqrt{(\cos x-1{)}^2-4})+C$
Hence, this is the answer.