NCERT Class 12 · Ch 7 Exercise 7.8 · Q17
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.8 Question 17
Evaluate \(\int\limits_{0}^{\frac{\pi}{4}} \left(x^{3} + 2 \sec^{2}{\left(x \right)} + 2\right)\, dx\).
💡 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: \(\frac{\pi^{4}}{1024} + \frac{\pi}{2} + 2\)
- 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)=\(\frac{x^{4}}{4} + 2 x + 2 \tan{\left(x \right)}\).
- At the upper and lower limits, F(b)=\(\frac{\pi^{4}}{1024} + \frac{\pi}{2} + 2\) and F(a)=\(0\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{\pi^{4}}{1024} + \frac{\pi}{2} + 2\).
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