Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #70
Evaluate \(\int \frac{x^{2}}{1 - x^{4}}\,dx\). Work on each connected open interval where the real integrand is defined.
✅ Step-by-Step Solution
Final Answer: \(\frac{\log{\left(\left|{\frac{x + 1}{x - 1}}\right| \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2}\) + C
- Factor 1−x⁴=(1−x²)(1+x²).
- The integrand equals (1/2)[1/(1−x²)−1/(1+x²)]. Also 1/(1−x²)=(1/2)[1/(1+x)+1/(1−x)].
- Integrate the reciprocal linear factors to (1/2)ln|(1+x)/(1−x)| and the quadratic to arctan x, retaining the outer halves.
- Therefore the required integral is \(\frac{\log{\left(\left|{\frac{x + 1}{x - 1}}\right| \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Factor 1−x⁴=(1−x²)(1+x²).
- [1 mark] The integrand equals (1/2)[1/(1−x²)−1/(1+x²)]. Also 1/(1−x²)=(1/2)[1/(1+x)+1/(1−x)].
- [1 mark] Integrate the reciprocal linear factors to (1/2)ln|(1+x)/(1−x)| and the quadratic to arctan x, retaining the outer halves.
- [0.5 marks] Therefore the required integral is \(\frac{\log{\left(\left|{\frac{x + 1}{x - 1}}\right| \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2}\) + C.
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