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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #158

Evaluate \(\int \frac{\sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 2 \cos{\left(x \right)} + 3}}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(- \operatorname{asin}{\left(\frac{\sqrt{5} \left(\cos{\left(x \right)} - 1\right)}{5} \right)}\) + C
  1. Set u=cos x−1, du=−sin x dx.
  2. Since sin²x+2cos x+3=5−(cos x−1)²=5−u², the integral becomes −∫du/√(5−u²).
  3. This is −arcsin(u/√5); substitute u=cos x−1.
  4. Therefore the required integral is \(- \operatorname{asin}{\left(\frac{\sqrt{5} \left(\cos{\left(x \right)} - 1\right)}{5} \right)}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Set u=cos x−1, du=−sin x dx.
  • [1 mark] Since sin²x+2cos x+3=5−(cos x−1)²=5−u², the integral becomes −∫du/√(5−u²).
  • [1 mark] This is −arcsin(u/√5); substitute u=cos x−1.
  • [0.5 marks] Therefore the required integral is \(- \operatorname{asin}{\left(\frac{\sqrt{5} \left(\cos{\left(x \right)} - 1\right)}{5} \right)}\) + C.

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