QuestionFind the integral: $\int x^{2} \ln (x) \, dx$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the integral of the function $x^{} \ln(x)$ with respect to $x$ . . We will use the method of integration by parts, which is based on the product rule of differentiation. The formula for integration by parts is $\int u dv = uv - \int v du$ .

STEP 2

First, we need to identify the functions $u$ and $dv$ in the integral. A good rule of thumb is to choose $u$ as the function that becomes simpler when differentiated. In this case, we choose $u = \ln(x)$ and $dv = x^2 dx$ .
$u = \ln(x)$ $dv = x^2 dx$

STEP 3

Now, we need to find the derivatives and integrals of $u$ and $dv$ respectively. The derivative of $u$ with respect to $x$ is $du = \frac{1}{x} dx$ and the integral of $dv$ is $v = \frac{x^3}{3}$ .
$du = \frac{1}{x} dx$ $v = \frac{x^3}{3}$

STEP 4

Substitute $u$ , $v$ , $du$ , and $dv$ into the integration by parts formula.
$\int x^{2} \ln(x) dx = uv - \int v du$

STEP 5

Substitute the values for $u$ , $v$ , $du$ , and $dv$ into the equation.
$\int x^{2} \ln(x) dx = \ln(x) \cdot \frac{x^3}{3} - \int \frac{x^3}{3} \cdot \frac{1}{x} dx$

STEP 6

implify the integral on the right side of the equation.
$\int x^{2} \ln(x) dx = \frac{x^3 \ln(x)}{3} - \int \frac{x^2}{3} dx$

STEP 7

Calculate the integral on the right side of the equation.
$\int x^{2} \ln(x) dx = \frac{x^3 \ln(x)}{3} - \frac{x^3}{9} + C$ Where $C$ is the constant of integration.
The integral of $x^{2} \ln(x)$ with respect to $x$ is $\frac{x^3 \ln(x)}{3} - \frac{x^3}{9} + C$ .

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