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JEE Main 2004
Exam
JEE Main
Year
2004
Subject
Mathematics
Chapter
Indefinite Integration

JEE Main 2004 · Mathematics · Indefinite Integration

If ∫ ( x)/( (x-α)) d x=A x+B (x-α)+C , then value of (A, B) is

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

Options

  1. A(sin⁡α,cos⁡α)(\sin \alpha, \cos \alpha)
  2. B(cos⁡α,sin⁡α)(\cos \alpha, \sin \alpha)
  3. C(−sin⁡α,cos⁡α)(-\sin \alpha, \cos \alpha)
  4. D(−cos⁡α,sin⁡α)(-\cos \alpha, \sin \alpha)

Correct answer

B. (cos⁡α,sin⁡α)(\cos \alpha, \sin \alpha)

Step-by-step solution

Put x−α=tx-\alpha=t ⇒∫sin⁡(α+t)sin⁡tdt=sin⁡α∫cot⁡tdt+cos⁡α∫dt\Rightarrow \int \frac{\sin (\alpha+t)}{\sin t} d t=\sin \alpha \int \cot t d t+\cos \alpha \int d t =cos⁡α(x−α)+sin⁡αln⁡∣sin⁡t∣+c=\cos \alpha(x-\alpha)+\sin \alpha \ln |\sin t|+c A=cos⁡α,B=sin⁡αA=\cos \alpha, B=\sin \alpha
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