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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #62

Evaluate \(\int \frac{x^{4}}{\left(x - 1\right) \left(x^{2} + 1\right)}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\frac{x^{2}}{2} + x - \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\log{\left(\left|{x - 1}\right| \right)}}{2} - \frac{\operatorname{atan}{\left(x \right)}}{2}\) + C
  1. Polynomial division gives x⁴/[(x−1)(x²+1)]=x+1+1/[(x−1)(x²+1)].
  2. The remaining fraction is 1/[2(x−1)]−(x+1)/[2(x²+1)].
  3. Integrate x+1 and each partial fraction: x²/2+x+(1/2)ln|x−1|−(1/4)ln(x²+1)−(1/2)arctan x.
  4. Therefore the required integral is \(\frac{x^{2}}{2} + x - \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\log{\left(\left|{x - 1}\right| \right)}}{2} - \frac{\operatorname{atan}{\left(x \right)}}{2}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Polynomial division gives x⁴/[(x−1)(x²+1)]=x+1+1/[(x−1)(x²+1)].
  • [1 mark] The remaining fraction is 1/[2(x−1)]−(x+1)/[2(x²+1)].
  • [1 mark] Integrate x+1 and each partial fraction: x²/2+x+(1/2)ln|x−1|−(1/4)ln(x²+1)−(1/2)arctan x.
  • [0.5 marks] Therefore the required integral is \(\frac{x^{2}}{2} + x - \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\log{\left(\left|{x - 1}\right| \right)}}{2} - \frac{\operatorname{atan}{\left(x \right)}}{2}\) + C.

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