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Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{\cos x}{\cos (x-a)} d x$

Solution:

Add and subtract a from $x$ in the numerator

$\therefore$ The equation becomes

$\Rightarrow \int \frac{\cos (x-a+a)}{\cos (x-a)}$

Numerator is of the form $\cos (A+B)=\cos A \cos B-\sin A \sin B$

Where $A=x-a ; B=a$

$\Rightarrow \int \frac{\cos (x-a) \cos a}{\cos (x-a)} d x-\int \frac{\sin (x-a) \sin a}{\cos (x-a)} d x$

$\Rightarrow \cos a \int d x-\sin a \int \tan (x-a) d x$

As $\int \tan x=\ln |\sec x|+c$

$\Rightarrow x \cos a-\sin a \frac{\ln |\sec (x-a)|}{(x-a)}+c$