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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #159

Evaluate \(\int \frac{x^{2}}{x^{4} + x^{2} - 2}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\frac{\log{\left(\left|{\frac{x - 1}{x + 1}}\right| \right)}}{6} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{3}\) + C
  1. Factor x⁴+x²−2=(x²+2)(x²−1).
  2. Partial fractions give x²/[(x²+2)(x²−1)]=2/[3(x²+2)]+1/[3(x²−1)].
  3. Integrate the first term by arctangent and the second using ∫dx/(x²−1)=(1/2)ln|(x−1)/(x+1)|.
  4. Therefore the required integral is \(\frac{\log{\left(\left|{\frac{x - 1}{x + 1}}\right| \right)}}{6} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{3}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Factor x⁴+x²−2=(x²+2)(x²−1).
  • [1 mark] Partial fractions give x²/[(x²+2)(x²−1)]=2/[3(x²+2)]+1/[3(x²−1)].
  • [1 mark] Integrate the first term by arctangent and the second using ∫dx/(x²−1)=(1/2)ln|(x−1)/(x+1)|.
  • [0.5 marks] Therefore the required integral is \(\frac{\log{\left(\left|{\frac{x - 1}{x + 1}}\right| \right)}}{6} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{3}\) + C.

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