Solve the integral. \int x \ln x dx

Question
Applications of integrals
asked 2021-02-24
Solve the integral.
\(\displaystyle\int{x}{\ln{{x}}}{\left.{d}{x}\right.}\)

Answers (1)

2021-02-25
Apply integration by parts:
\(\displaystyle\int{u}{d}{v}={u}{v}-\int{v}{d}{u}\)
Let, \(\displaystyle{u}={\ln{{x}}}\)
\(\displaystyle{v}={\frac{{{x}^{{{2}}}}}{{{2}}}}\)
v'=x
dv=xdx
\(\displaystyle\int{x}{\ln{{x}}}{\left.{d}{x}\right.}={\frac{{{1}}}{{{2}}}}{x}^{{{2}}}{\ln{{\left({x}\right)}}}-\int{\frac{{{x}}}{{{2}}}}{\left.{d}{x}\right.}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{x}^{{{2}}}{\ln{{\left({x}\right)}}}-{\frac{{{x}^{{{2}}}}}{{{4}}}}+{C}\)
0

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