Evaluate the following integrals : ∫sin(x-a)/sin(x-b) dx ​

Question

Evaluate the following integrals :

\(\int\frac{sin(x-a)}{sin(x-b)}\)dx

Answer 1

While solving these types of questions, 

It is better to eliminate the denominator.

\(\int\frac{sin(x-a)}{sin(x-b)}\)dx

Add and subtract b in (x - a),

⇒ \(\int\frac{sin(x-a+b-b)}{sin(x-b)}\)dx

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

Numerator is of the form sin (A + B) = sinA cosB + cosA sinB 

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