NCERT Class 12 · Ch 7 Exercise 7.3 · Q11
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.3 Question 11
Evaluate \(\int \cos^{4}{\left(2 x \right)}\, dx\).
💡 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: \(\frac{3 x}{8} + \frac{\sin{\left(4 x \right)}}{8} + \frac{\sin{\left(8 x \right)}}{64}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Use cos⁴u=(3+4cos 2u+cos 4u)/8.
- An antiderivative is F(x)=\(\frac{3 x}{8} + \frac{\sin{\left(4 x \right)}}{8} + \frac{\sin{\left(8 x \right)}}{64}\).
- Differentiate this expression to check that F′(x)=\(\cos^{4}{\left(2 x \right)}\). Add an arbitrary constant C on each domain component.
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