NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q35
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 35
Evaluate \(\int\limits_{0}^{\frac{\pi}{2}} \sin^{3}{\left(x \right)}\, dx\).
💡 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: \(\frac{2}{3}\)
- Work on each open interval where the original integrand is real and defined.
- Use sin³x=sin x(1−cos²x), then u=cos x.
- An antiderivative is F(x)=\(\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}\).
- At the upper and lower limits, F(b)=\(0\) and F(a)=\(- \frac{2}{3}\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{2}{3}\).
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