NCERT Class 12 · Ch 7 Exercise 7.8 · Q18
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.8 Question 18
Evaluate \(\int\limits_{0}^{\pi} \left(\sin^{2}{\left(\frac{x}{2} \right)} - \cos^{2}{\left(\frac{x}{2} \right)}\right)\, dx\).
💡 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: \(0\)
- Work on each open interval where the original integrand is real and defined.
- Use linearity and the elementary power, logarithm, trigonometric and exponential rules. Where a product x eˣ occurs use integration by parts; for rational terms split off the denominator derivative.
- An antiderivative is F(x)=\(- \sin{\left(x \right)}\).
- At the upper and lower limits, F(b)=\(0\) and F(a)=\(0\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(0\).
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