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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #71

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

✅ Step-by-Step Solution

Final Answer: \(- \frac{\cos{\left(2 x \right)}}{8} - \frac{\cos{\left(4 x \right)}}{16} + \frac{\cos{\left(6 x \right)}}{24}\) + C
  1. Use sin2x sin3x=(cos x−cos5x)/2.
  2. Then sin x sin2x sin3x=(sin2x−sin6x+sin4x)/4 by product-to-sum.
  3. Integrate each sine, using ∫sin(kx)dx=−cos(kx)/k.
  4. Therefore the required integral is \(- \frac{\cos{\left(2 x \right)}}{8} - \frac{\cos{\left(4 x \right)}}{16} + \frac{\cos{\left(6 x \right)}}{24}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Use sin2x sin3x=(cos x−cos5x)/2.
  • [1 mark] Then sin x sin2x sin3x=(sin2x−sin6x+sin4x)/4 by product-to-sum.
  • [1 mark] Integrate each sine, using ∫sin(kx)dx=−cos(kx)/k.
  • [0.5 marks] Therefore the required integral is \(- \frac{\cos{\left(2 x \right)}}{8} - \frac{\cos{\left(4 x \right)}}{16} + \frac{\cos{\left(6 x \right)}}{24}\) + C.

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