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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #60

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

✅ Step-by-Step Solution

Final Answer: \(\frac{\left(\sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{x}}{5}\) + C
  1. Let I=∫e^x sin2x dx and J=∫e^x cos2x dx. Integration by parts gives I=e^x sin2x−2J.
  2. Applying parts to J gives J=e^x cos2x+2I.
  3. Substitute into the first relation: 5I=e^x(sin2x−2cos2x), up to an arbitrary constant.
  4. Therefore the required integral is \(\frac{\left(\sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{x}}{5}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Let I=∫e^x sin2x dx and J=∫e^x cos2x dx. Integration by parts gives I=e^x sin2x−2J.
  • [1 mark] Applying parts to J gives J=e^x cos2x+2I.
  • [1 mark] Substitute into the first relation: 5I=e^x(sin2x−2cos2x), up to an arbitrary constant.
  • [0.5 marks] Therefore the required integral is \(\frac{\left(\sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{x}}{5}\) + C.

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