Question

$\int \frac{\cos x}{\cos \left(x-a\right)}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{\cos x}{\cos \left(x-a\right)}dx \\ \mathrm{Putting}x-a=t \\ \Rightarrow x=a+t \\ \Rightarrow dx=dt \\ \therefore I=\int \frac{\cos \left(a+t\right)dt}{\cos t} \\ =\int \frac{\cos a\cos t}{\cos t}-\frac{\sin a\sin t}{\cos t}dt \\ =\int \left(\cos a-\sin a\tan t\right)dt \\ =t\cos a-\sin a\ln \left|\sec t\right|+\mathrm{C} \\ =\left(x-a\right)\cos a-\sin a\ln \left|\sec \left(x-a\right)\right|+\mathrm{C}\left[\because t=x-a\right] \end{gathered}$$