NCERT Class 12 · Ch 7 Exercise 7.2 · Q13
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 13
Evaluate \(\int \frac{x^{2}}{\left(3 x^{3} + 2\right)^{3}}\, dx\).
💡 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: \(- \frac{1}{18 \left(3 x^{3} + 2\right)^{2}}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=2+3x³, du=9x²dx; integrate u^(−3)du/9.
- An antiderivative is F(x)=\(- \frac{1}{18 \left(3 x^{3} + 2\right)^{2}}\).
- Differentiate this expression to check that F′(x)=\(\frac{x^{2}}{\left(3 x^{3} + 2\right)^{3}}\). Add an arbitrary constant C on each domain component.
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