Question

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

Answer 1

$\begin{array}{c}{\int }^{}\frac{\cos x}{\sqrt{{\sin }^{2}x-2\sin x-3}}dx\\ Putting\sin x=tand\cos xdx=dt,weget\\ ={\int }^{}\frac{\cos x}{\sqrt{{\sin }^{2}x-2\sin x-3}}dx\\ ={\int }^{}\frac{dt}{\sqrt{t^2-2t-3}}\\ ={\int }^{}\frac{dt}{\sqrt{{\left(t-1\right)}^2-2^2}}\\ =\log \left|\left(t-1\right)+\sqrt{{\left(t-1\right)}^2-2^2}\right|+C\\ =\log \left|\left(\sin x-1\right)+\sqrt{{\sin }^{2}x-2\sin x-3}\right|+C\end{array}$