# Solve the following integral. \int_{1}^{e}x^{2}\ln x dx

Dottie Parra 2021-02-13 Answered
Solve the following integral. ${\int }_{1}^{e}{x}^{2}\mathrm{ln}xdx$
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SoosteethicU
Step 1
Given:
${\int }_{1}^{e}{x}^{2}\mathrm{ln}xdx$
Step 2
Solution:
${\int }_{1}^{e}{x}^{2}\mathrm{ln}\left(x\right)dx={\left[\frac{{x}^{3}}{3}\mathrm{ln}x\right]}_{1}^{e}-{\int }_{1}^{e}\frac{{x}^{3}}{3}dx$
$=\left(\frac{{e}^{3}}{3}-0\right)-{\left(\frac{{x}^{3}}{9}\right)}_{1}^{e}$
$⇒{\int }_{1}^{e}{x}^{2}\mathrm{ln}xdx=\frac{2{e}^{3}}{9}+\frac{1}{9}$