Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #150
Evaluate \(\int \frac{e^{x}}{\sqrt{e^{2 x} - 4 e^{x} - 5}}\,dx\). Work for x>ln5.
✅ Step-by-Step Solution
Final Answer: \(\log{\left(\sqrt{e^{2 x} - 4 e^{x} - 5} + e^{x} - 2 \right)}\) + C
- Let u=e^x, du=e^x dx.
- Complete the radicand u²−4u−5=(u−2)²−9. Its real positive-radicand domain is u>5, hence x>ln5.
- The standard primitive is ln|u−2+√[(u−2)²−9]|. Substitute u=e^x; its argument is positive here.
- Therefore the required integral is \(\log{\left(\sqrt{e^{2 x} - 4 e^{x} - 5} + e^{x} - 2 \right)}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let u=e^x, du=e^x dx.
- [1 mark] Complete the radicand u²−4u−5=(u−2)²−9. Its real positive-radicand domain is u>5, hence x>ln5.
- [1 mark] The standard primitive is ln|u−2+√[(u−2)²−9]|. Substitute u=e^x; its argument is positive here.
- [0.5 marks] Therefore the required integral is \(\log{\left(\sqrt{e^{2 x} - 4 e^{x} - 5} + e^{x} - 2 \right)}\) + C.
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