NCERT Class 12 · Ch 7 Exercise 7.6 · Q6
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 6
Evaluate \(\int x^{2} \log{\left(x \right)}\, dx\).
💡 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: \(\frac{x^{3} \log{\left(\left|{x}\right| \right)}}{3} - \frac{x^{3}}{9}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Choose u=\(\log{\left(x \right)}\) and v=\(\frac{x^{3}}{3}\), so dv=\(x^{2}\)dx and du=\(\frac{1}{x}\)dx.
- Apply ∫u dv=uv−∫v du. The remaining integrand v·u′ is \(\frac{x^{2}}{3}\). Integrate it and subtract from uv.
- An antiderivative is F(x)=\(\frac{x^{3} \log{\left(\left|{x}\right| \right)}}{3} - \frac{x^{3}}{9}\).
- Differentiate this expression to check that F′(x)=\(x^{2} \log{\left(x \right)}\). Add an arbitrary constant C on each domain component.
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