Question

integrate this function

$\int x \: log \: x \: dx$
​

Answer 1

$\LARGE\color{black}{\colorbox{#FF7968}{A}\colorbox{#4FB3F6}{N}\colorbox{#FEDD8E}{S}\colorbox{#FBBE2E}{W}\colorbox{#60D399}{E}\colorbox{#6D83F3}{R}}$

$\int x \: log \: x \: dx$

This can be integrated using integration by parts, using ILATE rule, logx be the first dunction and x be the second one

Thus,

$\sf \bf :\longmapsto I = \int x \: log \: x \: dx$

$\displaystyle \sf \bf :\longmapsto I = logx\int x.dx−\int \left[ \dfrac{d}{dx} logx \int x.dx\right]dx$

$\displaystyle \sf \bf :\longmapsto I=\dfrac{x^2logx}{2} +C−\int \left(\dfrac{1}{x} . \dfrac{x^2}{2}+C_2\right).dx$

$\displaystyle \sf \bf :\longmapsto I=\dfrac{x^2logx}{2}−\int\left[\dfrac{x}{2}+C_2\right]dx$

$\sf \bf :\longmapsto I=\dfrac{x^2logx}{2}−\dfrac{x^2}{4}+ C$

$\boxed{\sf \bf\int x \: log \: x \: dx =\dfrac{x^2logx}{2} - \dfrac{x^2}{2}+C}$

Formulas used,

$\boxed{\begin{array}{c|c|c} \sf \bf \int f(x)g(x) dx& =& \sf \bf f(x)\int g(x)dx-\int[f'(x)\int g(x)dx]dx \\ \sf \bf \dfrac{d}{dx}logx & = & \sf \bf\dfrac{1}{x}\\ \sf \bf \int x^n dx & = & \sf \bf\dfrac{x^{n+1}}{n+1}\end{array}}$