NCERT Class 12 · Ch 7 Exercise 7.3 · Q6
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.3 Question 6
Evaluate \(\int \sin{\left(x \right)} \sin{\left(2 x \right)} \sin{\left(3 x \right)}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ 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.
- Work on each open interval where the original integrand is real and defined.
- Product-to-sum gives [sin 2x+sin 4x−sin 6x]/4.
- An antiderivative is F(x)=\(- \frac{\cos{\left(2 x \right)}}{8} - \frac{\cos{\left(4 x \right)}}{16} + \frac{\cos{\left(6 x \right)}}{24}\).
- Differentiate this expression to check that F′(x)=\(\sin{\left(x \right)} \sin{\left(2 x \right)} \sin{\left(3 x \right)}\). Add an arbitrary constant C on each domain component.
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