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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #163

Evaluate \(\int e^{2 x} \sin{\left(3 x + 1 \right)}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\frac{\left(2 \sin{\left(3 x + 1 \right)} - 3 \cos{\left(3 x + 1 \right)}\right) e^{2 x}}{13}\) + C
  1. Let I=∫e^(2x)sin(3x+1)dx and J=∫e^(2x)cos(3x+1)dx. Parts gives I=e^(2x)sin(3x+1)/2−3J/2.
  2. A second integration by parts gives J=e^(2x)cos(3x+1)/2+3I/2.
  3. Eliminate J: (13/4)I=e^(2x)[sin(3x+1)/2−3cos(3x+1)/4], yielding the stated primitive.
  4. Therefore the required integral is \(\frac{\left(2 \sin{\left(3 x + 1 \right)} - 3 \cos{\left(3 x + 1 \right)}\right) e^{2 x}}{13}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Let I=∫e^(2x)sin(3x+1)dx and J=∫e^(2x)cos(3x+1)dx. Parts gives I=e^(2x)sin(3x+1)/2−3J/2.
  • [1 mark] A second integration by parts gives J=e^(2x)cos(3x+1)/2+3I/2.
  • [1 mark] Eliminate J: (13/4)I=e^(2x)[sin(3x+1)/2−3cos(3x+1)/4], yielding the stated primitive.
  • [0.5 marks] Therefore the required integral is \(\frac{\left(2 \sin{\left(3 x + 1 \right)} - 3 \cos{\left(3 x + 1 \right)}\right) e^{2 x}}{13}\) + C.

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