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Step-by-step Solution

Solve the trigonometric integral $\int x^2\ln\left(x\right)dx$

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y
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e
π
ln
log
log
lim
d/dx
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sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
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sinh
cosh
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sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Answer

$x^{3}\left(-\frac{1}{3}+\frac{1}{3}\ln\left(x\right)\right)+\frac{2}{9}x^{3}+C_0$

Step-by-step explanation

Problem to solve:

$\int x^2\:\ln\:x\:dx$
1

Use the integration by parts theorem to calculate the integral $\int x^2\ln\left(x\right)dx$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x^2}\\ \displaystyle{du=2xdx}\end{matrix}$

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Answer

$x^{3}\left(-\frac{1}{3}+\frac{1}{3}\ln\left(x\right)\right)+\frac{2}{9}x^{3}+C_0$