NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 14

✅ Step-by-Step Solution

1. Work on each open interval where the original integrand is real and defined.
2. Choose u=\(\log{\left(x \right)}^{2}\) and v=\(\frac{x^{2}}{2}\), so dv=\(x\)dx and du=\(\frac{2 \log{\left(x \right)}}{x}\)dx.
3. Apply ∫u dv=uv−∫v du. The remaining integrand v·u′ is \(x \log{\left(x \right)}\). Integrate it and subtract from uv.
4. An antiderivative is F(x)=\(\frac{x^{2} \log{\left(\left|{x}\right| \right)}^{2}}{2} - \frac{x^{2} \log{\left(\left|{x}\right| \right)}}{2} + \frac{x^{2}}{4}\).
5. Differentiate this expression to check that F′(x)=\(x \log{\left(x \right)}^{2}\). Add an arbitrary constant C on each domain component.