Question

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

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{\sin x}{\sqrt{{\cos }^{2}x-2\cos x-3}}dx \\ \mathrm{Putting}\cos x=t \\ \Rightarrow -\sin xdx=dt \\ \Rightarrow \sin xdx=-dt \\ \therefore I=-\int \frac{dt}{\sqrt{t^2-2t-3}} \\ =-\int \frac{dt}{\sqrt{t^2-2t+1-4}} \\ =-\int \frac{dt}{\sqrt{{\left(t-1\right)}^2-{\left(2\right)}^2}} \\ =-\ln \left|t-1+\sqrt{{\left(t-1\right)}^2-4}\right|+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left[\because \int \frac{\mathit{1}}{\sqrt{x^{\mathit{2}}\mathit{-}{a}^{\mathit{2}}}}dx\mathit{=}\ln \left|x+\sqrt{x^{\mathit{2}}\mathit{-}{a}^{\mathit{2}}}\right|+C\right] \\ =-\ln \left|\left(\cos x-1\right)+\sqrt{{\cos }^{2}x-2\cos x-3}\right|+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left[\because t=\cos \mathit{}x\right] \end{gathered}$$