Question

Evaluate: $\int \frac{\cos x}{\cos (x-a)}dx$

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

Answer 1

The correct option is C $(x-a)\cos a-\sin a\log \sec (x-a)$

Let $I=\int \frac{\cos x}{\cos (x-a)}dx$
Put $x-a=t\Rightarrow x=a+t$ and $dx=dt$
Therefore
$I=\int \frac{\cos (a+t)}{\cos t}dt=\int \frac{\cos a\cos t-\sin a\sin t}{\cos t}dt$
$=\int (\cos a-\sin a\tan t)dt=t\cos a-\sin a\log \sec t$
$=(x-a)\cos a-\sin a\log \sec (x-a)$
Hence, option 'C' is correct.