NCERT Class 12 · Ch 7 Exercise 7.8 · Q20
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.8 Question 20
Evaluate \(\int\limits_{0}^{1} \left(x e^{x} + \sin{\left(\frac{\pi x}{4} \right)}\right)\, dx\).
💡 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: \(\frac{- 2 \sqrt{2} + \pi + 4}{\pi}\)
- 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)=\(\left(x - 1\right) e^{x} - \frac{4 \cos{\left(\frac{\pi x}{4} \right)}}{\pi}\).
- At the upper and lower limits, F(b)=\(- \frac{2 \sqrt{2}}{\pi}\) and F(a)=\(- \frac{4}{\pi} - 1\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{- 2 \sqrt{2} + \pi + 4}{\pi}\).
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