Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #103
Evaluate \(\int_0^\pi e^{2x}\sin(x+\pi/4)dx\).
✅ Step-by-Step Solution
Final Answer: −(e^(2π)+1)/(5√2)
- Let I=∫e^(2x)sin(x+π/4)dx and J=∫e^(2x)cos(x+π/4)dx. Parts gives I=e^(2x)sin(x+π/4)/2−J/2.
- A second integration by parts gives J=e^(2x)cos(x+π/4)/2+I/2; eliminate J to obtain F=e^(2x)[2sin(x+π/4)−cos(x+π/4)]/5.
- Evaluate F(π)=−e^(2π)/(5√2) and F(0)=1/(5√2), then subtract.
- Therefore the integral equals −(e^(2π)+1)/(5√2).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let I=∫e^(2x)sin(x+π/4)dx and J=∫e^(2x)cos(x+π/4)dx. Parts gives I=e^(2x)sin(x+π/4)/2−J/2.
- [1 mark] A second integration by parts gives J=e^(2x)cos(x+π/4)/2+I/2; eliminate J to obtain F=e^(2x)[2sin(x+π/4)−cos(x+π/4)]/5.
- [1 mark] Evaluate F(π)=−e^(2π)/(5√2) and F(0)=1/(5√2), then subtract.
- [0.5 marks] Therefore the integral equals −(e^(2π)+1)/(5√2).
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