# Integral of x^2ln(x)

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

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

## Step by step solution

Problem

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

$\begin{matrix}\displaystyle{u=\ln\left(x\right)}\\ \displaystyle{du=\frac{1}{x}dx}\end{matrix}$
3

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=x^2dx}\\ \displaystyle{\int dv=\int x^2dx}\end{matrix}$
4

Solve the integral

$v=\int x^2dx$
5

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{x^{3}}{3}$
6

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{x^{3}}{3}\ln\left(x\right)-\int\frac{1}{x}\cdot\frac{x^{3}}{3}dx$
7

Multiplying fractions

$\frac{x^{3}}{3}\ln\left(x\right)-\int\frac{x^{3}}{3x}dx$
8

Simplifying the fraction by $x$

$\frac{x^{3}}{3}\ln\left(x\right)-\int\frac{x^{2}}{3}dx$
9

Taking the constant out of the integral

$\frac{x^{3}}{3}\ln\left(x\right)-1\cdot \frac{1}{3}\int x^{2}dx$
10

Multiply $\frac{1}{3}$ times $-1$

$\frac{x^{3}}{3}\ln\left(x\right)-\frac{1}{3}\int x^{2}dx$
11

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{x^{3}}{3}\ln\left(x\right)-\frac{1}{3}\cdot\frac{x^{3}}{3}$
12

Simplify the fraction

$\frac{x^{3}}{3}\ln\left(x\right)-\frac{1}{9}x^{3}$
13

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

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

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### Main topic:

Integration by parts

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