NCERT Class 12 · Ch 7 Exercise 7.4 · Q6
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 6
Evaluate \(\int \frac{x^{2}}{1 - x^{6}}\, dx\).
💡 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: \(\frac{\log{\left(\left|{\frac{x^{3} + 1}{x^{3} - 1}}\right| \right)}}{6}\) + C.
- Work on each open interval where the integrand is real and defined.
- u=x³, du=3x²dx; partial fractions 1/(1−u²).
- An antiderivative is F(x)=\(\frac{\log{\left(\left|{\frac{x^{3} + 1}{x^{3} - 1}}\right| \right)}}{6}\).
- Differentiate this expression to check that F′(x)=\(\frac{x^{2}}{1 - x^{6}}\). Add an arbitrary constant C on each domain component.
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