Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #74
Evaluate \(\int \frac{e^{x}}{\sqrt{- e^{2 x} - 4 e^{x} + 5}}\,dx\). Work for x<0, the real positive-radicand domain.
✅ Step-by-Step Solution
Final Answer: \(\operatorname{asin}{\left(\frac{e^{x}}{3} + \frac{2}{3} \right)}\) + C
- Set u=e^x, du=e^x dx.
- Complete the square: 5−4u−u²=9−(u+2)². The integral is ∫du/√(9−(u+2)²).
- Use v=(u+2)/3 to obtain arcsin v, then substitute u=e^x.
- Therefore the required integral is \(\operatorname{asin}{\left(\frac{e^{x}}{3} + \frac{2}{3} \right)}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Set u=e^x, du=e^x dx.
- [1 mark] Complete the square: 5−4u−u²=9−(u+2)². The integral is ∫du/√(9−(u+2)²).
- [1 mark] Use v=(u+2)/3 to obtain arcsin v, then substitute u=e^x.
- [0.5 marks] Therefore the required integral is \(\operatorname{asin}{\left(\frac{e^{x}}{3} + \frac{2}{3} \right)}\) + C.
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