Question

Evaluate $\int \frac{\cos x}{\cos (x-a)}dx$
(where $C$ is constant of integration)

• A. $x\cos a-\sin a\ln |\sec (x-a)|+C$
• B. $x\cos a+\sin a\ln |\sec (x-a)|+C$
• C. $-x\cos a-\sin a\ln |\sec (x-a)|+C$
• D. $-x\cos a+\sin a\ln |\sec (x-a)|+C$

Answer 1

The correct option is A $x\cos a-\sin a\ln |\sec (x-a)|+C$

Let $I=\int \frac{\cos \left(x\right)}{\cos (x-a)}dx$
$I=\int \frac{\cos (x-a+a)}{\cos (x-a)}dx$

$\Rightarrow I=\int \frac{\cos (x-a)\cos a-\sin (x-a)\sin a}{\cos (x-a)}dx$

$\Rightarrow I=\int (\cos a-\tan (x-a\left)\sin a\right)dx$

$\therefore I=x\cos a-\sin a\ln |\sec (x-a)|+C$