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Question: If \(\int_{}^{}\frac{\sin x}{\sin(x - \alpha)}\)dx = Ax + B log sin (x – a) + c then value of (A, B)...

If ∫sin⁡xsin⁡(x−α)\int_{}^{}\frac{\sin x}{\sin(x - \alpha)}dx = Ax + B log sin (x – a) + c then value of (A, B) is

A

(– sin a, cos a)

B

(cos a, sin a)

C

(sin a, cos a)

D

(– cos a, sin a)

Answer

(cos a, sin a)

Explanation

Solution

∫sin⁡(x−α+α)sin⁡(x−α)\int \frac { \sin ( x - \alpha + \alpha ) } { \sin ( x - \alpha ) }dx

= ∫sin⁡(x−α)cos⁡α+cos⁡(x−α)sin⁡αsin⁡(x−α)\int \frac { \sin ( x - \alpha ) \cos \alpha + \cos ( x - \alpha ) \sin \alpha } { \sin ( x - \alpha ) } dx

= cos a ∫dx+sin⁡α∫cot⁡(x−α)dx\int_{}^{}{dx + \sin\alpha}\int_{}^{}{\cot(x - \alpha)dx}= (cos a). x + sin a.

log {sin (x – a)} + c