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Variable Separable Method

If ∫sin xsin ...

Question

If $\int \frac{sin x}{sin (x - α)} d x = A x + B ln | sin (x - α) | + C,$ then value of $(A, B)$ is
(where $A, B$ are fixed constants and $C$ is constant of integration)

(- cos α, sin α)

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(cos α, sin α)

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(- sin α, cos α)

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(sin α, cos α)

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Solution

The correct option is B $(cos α, sin α)$
$\int \frac{sin x}{sin (x - α)} d x$
$= \int \frac{sin (x - α + α)}{sin (x - α)} d x$
$= \int \frac{sin (x - α) cos α + cos (x - α) sin α}{sin (x - α)} d x$
$= \int {cos α + sin α cot (x - α)} d x$
$= (cos α) x + (sin α) ln | sin (x - α) | + C$
Comparing with $A x + B ln | sin (x - α) | + C$ , we have
$A = cos α, B = sin α$

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