Integrals — Maths STD 12 Science — Question

$= \int {\left( {\log x} \right){x^2}dx} $

$= \log x\int {{x^2}dx - \int {\left( {\frac{d}{{dx}}\log x\int {{x^2}dx} } \right)dx} } $

[Applying product rule]

$= \left( {\log x} \right)\frac{{{x^3}}}{3} - \int {\frac{1}{x}.\frac{{{x^3}}}{3}dx} $

$= \frac{{{x^3}}}{3}\log x - \frac{1}{3}\int{x^2}dx$

$= \frac{{{x^3}}}{3}\log x - \frac{1}{3}\frac{{{x^3}}}{3} + c$

$= \frac{{{x^3}}}{3}\log x - \frac{{{x^3}}}{9} + c$