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Class 12 Integrals · Short Answer2 Marksindefinite_integration

Class 12 Maths · Integrals · Short Answer #58

Evaluate \(\int \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}}{\sin^{2}{\left(x \right)}}\,dx\). Work on intervals with sin x≠0.

✅ Step-by-Step Solution

Final Answer: \(- \frac{e^{x}}{\sin{\left(x \right)}}\) + C
  1. Rewrite (cos x−sin x)csc²x=csc x cot x−csc x.
  2. Differentiate −e^x csc x: it is e^x(csc x cot x−csc x), exactly the integrand.
  3. Therefore the required integral is \(- \frac{e^{x}}{\sin{\left(x \right)}}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Rewrite (cos x−sin x)csc²x=csc x cot x−csc x.
  • [1 mark] Differentiate −e^x csc x: it is e^x(csc x cot x−csc x), exactly the integrand.
  • [0.5 marks] Therefore the required integral is \(- \frac{e^{x}}{\sin{\left(x \right)}}\) + C.

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