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Class 12 Integrals · Short Answer2 Marksindefinite_integration

Class 12 Maths · Integrals · Short Answer #59

Evaluate \(\int \frac{- \cos{\left(2 a \right)} + \cos{\left(2 x \right)}}{- \cos{\left(a \right)} + \cos{\left(x \right)}}\,dx\). Work on intervals with cos x≠cos a; a is a real constant.

✅ Step-by-Step Solution

Final Answer: \(2 x \cos{\left(a \right)} + 2 \sin{\left(x \right)}\) + C
  1. Use cos2u=2cos²u−1 to factor the numerator as 2(cos x−cos a)(cos x+cos a).
  2. On the original domain, cancel cos x−cos a. Integrate 2(cos x+cos a) termwise.
  3. Therefore the required integral is \(2 x \cos{\left(a \right)} + 2 \sin{\left(x \right)}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Use cos2u=2cos²u−1 to factor the numerator as 2(cos x−cos a)(cos x+cos a).
  • [1 mark] On the original domain, cancel cos x−cos a. Integrate 2(cos x+cos a) termwise.
  • [0.5 marks] Therefore the required integral is \(2 x \cos{\left(a \right)} + 2 \sin{\left(x \right)}\) + C.

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