Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #79
Evaluate \(\int \frac{x^{2}}{x^{4} - x^{2} - 12}\,dx\). Work on each connected open interval where the real integrand is defined.
✅ Step-by-Step Solution
Final Answer: \(\frac{\log{\left(\left|{\frac{x - 2}{x + 2}}\right| \right)}}{7} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}}{7}\) + C
- Factor x⁴−x²−12=(x²−4)(x²+3).
- Partial fractions give x²/[(x²−4)(x²+3)]=4/[7(x²−4)]+3/[7(x²+3)].
- Use ∫dx/(x²−4)=(1/4)ln|(x−2)/(x+2)| and ∫dx/(x²+3)=arctan(x/√3)/√3.
- Therefore the required integral is \(\frac{\log{\left(\left|{\frac{x - 2}{x + 2}}\right| \right)}}{7} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}}{7}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Factor x⁴−x²−12=(x²−4)(x²+3).
- [1 mark] Partial fractions give x²/[(x²−4)(x²+3)]=4/[7(x²−4)]+3/[7(x²+3)].
- [1 mark] Use ∫dx/(x²−4)=(1/4)ln|(x−2)/(x+2)| and ∫dx/(x²+3)=arctan(x/√3)/√3.
- [0.5 marks] Therefore the required integral is \(\frac{\log{\left(\left|{\frac{x - 2}{x + 2}}\right| \right)}}{7} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}}{7}\) + C.
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