Evaluate the following integrals:

While solving these types of questions, it is better to eliminate the denominator.

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

Add and subtract $b$ in $(x-a)$

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

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

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

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

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

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

$\Rightarrow \int \cos (b-a) d x+\int \cot (x-b) \sin (b-a) d x$

$\Rightarrow \cos (b-a) \int d x+\sin (b-a) \int \cot (x-b) d x$

As $\int \cot (\mathrm{x}) \mathrm{dx}=\ln |\sin \mathrm{x}|$

$\Rightarrow \cos (b-a) x+\sin (b-a) \ln |\sin (x-b)|$