Verified Solution • 2025–26 Session moderate

QJEEM04Ma7f67: JEE Main 2004 Mathematics - Class 12 JEEM04Ma7f67 Solution: If then value of is

If ∫sin⁡xsin⁡(x−α)dx=Ax+Blog⁡sin⁡(x−α),+C,\int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}dx = Ax + B\log \sin \left( {x - \alpha } \right), + C,} then value of
(A,B)(A, B) is

Multiple Choice Options

  • A (−cos⁡α,sin⁡α)\left( { - \cos \alpha ,\sin \alpha } \right)
  • B (cos⁡α,sin⁡α)\left( { \cos \alpha ,\sin \alpha } \right)
  • C (−sin⁡α,cos⁡α)\left( { - \sin \alpha ,\cos \alpha } \right)
  • D (sin⁡α,cos⁡α)\left( { \sin \alpha ,\cos \alpha } \right)

Step-by-Step Verified Solution

∫sin⁡xsin⁡(x−α)dx\int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}} dx

=∫sin⁡(x−α+α)sin⁡(x−α)dx= \int {{{\sin \left( {x - \alpha + \alpha } \right)} \over {\sin \left( {x - \alpha } \right)}}} dx

=∫sin⁡(x−α)cos⁡α+cos⁡(x−α)sin⁡αsin⁡(x−α)= \int {{{\sin \left( {x - \alpha } \right)\cos \alpha + \cos \left( {x - \alpha } \right)\sin \alpha } \over {\sin \left( {x - \alpha } \right)}}}

=∫{cos⁡α+sin⁡α cot⁡(x−α)}dx= \int {\left\{ {\cos \alpha + \sin \alpha \,\cot \left. {\left( {x - \alpha } \right)} \right\}} \right.} dx

=(cos⁡α)x+(sin⁡α)log⁡ sin⁡(x−α)+C= \left( {\cos \alpha } \right)x + \left( {\sin \alpha } \right)\log \,\sin \left( {x - \alpha } \right) + C

∴\therefore A=cos⁡α,A = \cos \alpha , B=sin⁡αB = \sin \alpha

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