Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #90
Evaluate \(\int_{\ln\sqrt2}^{\ln\sqrt3}\frac{dx}{(e^x+e^{-x})(e^x-e^{-x})}\).
✅ Step-by-Step Solution
Final Answer: (1/4)ln(3/2)
- Multiply numerator and denominator by e^(2x), then set u=e^(2x), du=2e^(2x)dx. The limits become 2,3.
- The integral is (1/2)∫₂³du/(u²−1)=(1/4)[ln|(u−1)/(u+1)|]₂³.
- The endpoint ratio is [(3−1)/(3+1)]/[(2−1)/(2+1)]=(1/2)/(1/3)=3/2.
- Therefore the integral equals (1/4)ln(3/2).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Multiply numerator and denominator by e^(2x), then set u=e^(2x), du=2e^(2x)dx. The limits become 2,3.
- [1 mark] The integral is (1/2)∫₂³du/(u²−1)=(1/4)[ln|(u−1)/(u+1)|]₂³.
- [1 mark] The endpoint ratio is [(3−1)/(3+1)]/[(2−1)/(2+1)]=(1/2)/(1/3)=3/2.
- [0.5 marks] Therefore the integral equals (1/4)ln(3/2).
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