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Step-by-step Solution

Trigonometric integral $\int x^2\ln\left(x\right)dx$

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ln
log
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sin
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asin
acos
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sinh
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asinh
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Answer

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

Step-by-step explanation

Problem to solve:

$\int\left(x^2\ln\left(x\right)\right)dx$
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

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