Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #69
Evaluate \(\int x^{2} \log{\left(x^{2} + 1 \right)}\,dx\). Work on each connected open interval where the real integrand is defined.
✅ Step-by-Step Solution
Final Answer: \(\frac{x^{3} \log{\left(x^{2} + 1 \right)}}{3} - \frac{2 x^{3}}{9} + \frac{2 x}{3} - \frac{2 \operatorname{atan}{\left(x \right)}}{3}\) + C
- Use parts with u=ln(1+x²),dv=x²dx. Then I=x³ln(1+x²)/3−(2/3)∫x⁴/(1+x²)dx.
- Polynomial division gives x⁴/(1+x²)=x²−1+1/(1+x²).
- Integrating the remainder gives x³/3−x+arctan x; combine with the first term.
- Therefore the required integral is \(\frac{x^{3} \log{\left(x^{2} + 1 \right)}}{3} - \frac{2 x^{3}}{9} + \frac{2 x}{3} - \frac{2 \operatorname{atan}{\left(x \right)}}{3}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Use parts with u=ln(1+x²),dv=x²dx. Then I=x³ln(1+x²)/3−(2/3)∫x⁴/(1+x²)dx.
- [1 mark] Polynomial division gives x⁴/(1+x²)=x²−1+1/(1+x²).
- [1 mark] Integrating the remainder gives x³/3−x+arctan x; combine with the first term.
- [0.5 marks] Therefore the required integral is \(\frac{x^{3} \log{\left(x^{2} + 1 \right)}}{3} - \frac{2 x^{3}}{9} + \frac{2 x}{3} - \frac{2 \operatorname{atan}{\left(x \right)}}{3}\) + C.
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