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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #75

Evaluate \(\int \frac{\left(\sin{\left(4 x \right)} - 4\right) e^{x}}{1 - \cos{\left(4 x \right)}}\,dx\). Work on intervals with sin2x≠0.

✅ Step-by-Step Solution

Final Answer: \(e^{x} \cot{\left(2 x \right)}\) + C
  1. Use sin4x=2sin2x cos2x and 1−cos4x=2sin²2x.
  2. The integrand becomes e^x[cot2x−2csc²2x].
  3. Because (cot2x)′=−2csc²2x, this is the derivative of e^x cot2x.
  4. Therefore the required integral is \(e^{x} \cot{\left(2 x \right)}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Use sin4x=2sin2x cos2x and 1−cos4x=2sin²2x.
  • [1 mark] The integrand becomes e^x[cot2x−2csc²2x].
  • [1 mark] Because (cot2x)′=−2csc²2x, this is the derivative of e^x cot2x.
  • [0.5 marks] Therefore the required integral is \(e^{x} \cot{\left(2 x \right)}\) + C.

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