Evaluate : ∫sinx/sin (x+a) dx

Question

Evaluate : \(\int\frac{sin\,x}{sin(x+a)}\,dx\)

Answer 1

\(I = \int \frac{\sin x}{\sin(x + a)}dx\)

Let x + a = y

⇒ dx = dy

Substituting in the given integral, we get

\(I = \int \frac{\sin (y- a)}{\sin y\,dy}\)

⇒ \(\sin (y -a) = \sin y \times \cos a+ \cos y\times \sin a\)

\(I = \cos a \int dy - \sin a\int \cot y\,dy\)

⇒ \(I = (\cos a)y - (sin a) \log\sin y\)

⇒ \(I = (\cos a)(x + a) - (\sin a) \log[\sin (x + a) + c]\)

Answer 2

Let x + a = t

⇒ dx = dt