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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #61

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

✅ Step-by-Step Solution

Final Answer: \(\log{\left(\frac{x^{2} + 1}{x^{2} + 2} \right)}\) + C
  1. Set u=x², du=2x dx; the integral becomes ∫du/[(u+1)(u+2)].
  2. Partial fractions give 1/[(u+1)(u+2)]=1/(u+1)−1/(u+2).
  3. Integrate to ln|u+1|−ln|u+2| and substitute u=x². Both log arguments x²+1,x²+2 are positive.
  4. Therefore the required integral is \(\log{\left(\frac{x^{2} + 1}{x^{2} + 2} \right)}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Set u=x², du=2x dx; the integral becomes ∫du/[(u+1)(u+2)].
  • [1 mark] Partial fractions give 1/[(u+1)(u+2)]=1/(u+1)−1/(u+2).
  • [1 mark] Integrate to ln|u+1|−ln|u+2| and substitute u=x². Both log arguments x²+1,x²+2 are positive.
  • [0.5 marks] Therefore the required integral is \(\log{\left(\frac{x^{2} + 1}{x^{2} + 2} \right)}\) + C.

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