NCERT Class 12 · Ch 7 Exercise 7.9 · Q9
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.9 Question 9
Evaluate \(\int\limits_{\frac{1}{3}}^{1} \frac{\sqrt[3]{- x^{3} + x}}{x^{4}}\, dx\).
💡 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: A: \(6\)
- Work on each open interval where the original integrand is real and defined.
- Let u=1/x²−1, du=−2dx/x³. The integrand becomes −u^(1/3)du/2 and the new limits are 8 and 0.
- An antiderivative is F(x)=\(- \frac{3 \left(-1 + \frac{1}{x^{2}}\right)^{\frac{4}{3}}}{8}\).
- At the upper and lower limits, F(b)=\(0\) and F(a)=\(-6\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(6\).
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