NCERT Class 12 · Ch 7 Exercise 7.9 · Q2
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.9 Question 2
Evaluate \(\int\limits_{0}^{\frac{\pi}{2}} \sqrt{\sin{\left(x \right)}} \cos^{5}{\left(x \right)}\, dx\).
💡 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: \(\frac{64}{231}\)
- Work on each open interval where the original integrand is real and defined.
- u=sin x; cos⁴x=(1−u²)². Integrate u^(1/2)(1−2u²+u⁴).
- An antiderivative is F(x)=\(\frac{2 \sin^{\frac{11}{2}}{\left(x \right)}}{11} - \frac{4 \sin^{\frac{7}{2}}{\left(x \right)}}{7} + \frac{2 \sin^{\frac{3}{2}}{\left(x \right)}}{3}\).
- At the upper and lower limits, F(b)=\(\frac{64}{231}\) and F(a)=\(0\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{64}{231}\).
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