Question

If $ \int \frac{sin\,x}{sin\,\left(x-\alpha\right)}\, dx = Ax + Blogsin\left(x-\alpha\right) + C$, then value of $\left(A, B\right)$ is

• A. $(sin\,\alpha, \,cos\,\alpha)$
• B. $(cos \alpha, \,sin\, \alpha)$
• C. $(- sin\, \alpha\, cos\, \alpha)$
• D. $(- cos \alpha, \,sin\, \alpha)$

Answer 1

Answer: (B)

Put $x - α = t$
$\Rightarrow \int \frac{sin\left(\alpha+ t\right)}{sin\,t}\, dt \,sin\,\alpha \int\, cot\, tdt + cos\,\alpha\,\int\,dt$
$= cos\, α\left(x − α\right) + sin\, α\, ln \left|sin\, t \right|+c$
$A = cos\,α,\, B = sin\,α$