NCERT Class 12 · Ch 7 Exercise 7.2 · Q12
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 12
Evaluate \(\int x^{5} \sqrt[3]{x^{3} - 1}\, dx\).
💡 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: \(\frac{\left(x^{3} - 1\right)^{\frac{7}{3}}}{7} + \frac{\left(x^{3} - 1\right)^{\frac{4}{3}}}{4}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=x³−1, du=3x²dx and x³=u+1; integrate (u+1)u^(1/3)/3. For negative u, all thirds mean real cube-root powers.
- An antiderivative is F(x)=\(\frac{\left(x^{3} - 1\right)^{\frac{7}{3}}}{7} + \frac{\left(x^{3} - 1\right)^{\frac{4}{3}}}{4}\).
- Differentiate this expression to check that F′(x)=\(x^{5} \sqrt[3]{x^{3} - 1}\). Add an arbitrary constant C on each domain component.
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