Integrals — Maths STD 12 Science — Question

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

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

[Applying product rule]

$= \left( {\log 2x} \right)\frac{{{x^2}}}{2} - \int {\frac{1}{{2x}}.2.\frac{{{x^2}}}{2}dx} $

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

$= \frac{1}{2}{x^2}\log 2x - \frac{1}{2}\frac{{{x^2}}}{2} + c$

$= \frac{{{x^2}}}{2}\log 2x - \frac{{{x^2}}}{4} + c$