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Question: If \(\int_{}^{}\frac{\cos x}{\sin(x–\alpha)}\) = A log sin (x – a) – Bx + C then (A, B) =...

If ∫cos⁡xsin⁡(x–α)\int_{}^{}\frac{\cos x}{\sin(x–\alpha)} = A log sin (x – a) – Bx + C then (A, B) =

A

(cosa, sin a)

B

(sin a, cos a)

C

(– cos a, sin a)

D

( – sin a, cos a)

Answer

(cosa, sin a)

Explanation

Solution

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

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

= cos⁡α∫cot⁡(x−α)dx−sin⁡α∫dx\cos \alpha \int \cot ( x - \alpha ) d x - \sin \alpha \int d x