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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #165

Evaluate \(\int \frac{\sin{\left(2 x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right) \left(\sin^{2}{\left(x \right)} + 3\right)}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\frac{\log{\left(\sin^{2}{\left(x \right)} + 1 \right)}}{2} - \frac{\log{\left(\sin^{2}{\left(x \right)} + 3 \right)}}{2}\) + C
  1. Set u=sin²x, du=sin2x dx.
  2. The integral is ∫du/[(u+1)(u+3)]=(1/2)∫[1/(u+1)−1/(u+3)]du.
  3. Integrate and substitute u=sin²x; both logarithm arguments are positive.
  4. Therefore the required integral is \(\frac{\log{\left(\sin^{2}{\left(x \right)} + 1 \right)}}{2} - \frac{\log{\left(\sin^{2}{\left(x \right)} + 3 \right)}}{2}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Set u=sin²x, du=sin2x dx.
  • [1 mark] The integral is ∫du/[(u+1)(u+3)]=(1/2)∫[1/(u+1)−1/(u+3)]du.
  • [1 mark] Integrate and substitute u=sin²x; both logarithm arguments are positive.
  • [0.5 marks] Therefore the required integral is \(\frac{\log{\left(\sin^{2}{\left(x \right)} + 1 \right)}}{2} - \frac{\log{\left(\sin^{2}{\left(x \right)} + 3 \right)}}{2}\) + C.

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