Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #89
Evaluate \(\int_0^{\pi/2}e^x\sin x\,dx\).
✅ Step-by-Step Solution
Final Answer: (e^(π/2)+1)/2
- Let I=∫e^x sin x dx and J=∫e^x cos x dx. Parts gives I=e^x sin x−J and J=e^x cos x+I.
- Eliminating J yields the primitive F=e^x(sin x−cos x)/2.
- At π/2, F=e^(π/2)/2; at 0, F=−1/2. Their difference is (e^(π/2)+1)/2.
- Therefore the integral equals (e^(π/2)+1)/2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let I=∫e^x sin x dx and J=∫e^x cos x dx. Parts gives I=e^x sin x−J and J=e^x cos x+I.
- [1 mark] Eliminating J yields the primitive F=e^x(sin x−cos x)/2.
- [1 mark] At π/2, F=e^(π/2)/2; at 0, F=−1/2. Their difference is (e^(π/2)+1)/2.
- [0.5 marks] Therefore the integral equals (e^(π/2)+1)/2.
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