Calculus Examples

$\int x \ln (x) x d x$

Step 1

Simplify.

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Step 1.1

Raise $x$ to the power of $1$ .

$\int x^{1} x \ln (x) d x$

Step 1.2

Raise $x$ to the power of $1$ .

$\int x^{1} x^{1} \ln (x) d x$

Step 1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{1 + 1} \ln (x) d x$

Step 1.4

Add $1$ and $1$ .

$\int x^{2} \ln (x) d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{2}$ .

$\ln (x) (\frac{1}{3} x^{3}) - \int \frac{1}{3} x^{3} \frac{1}{x} d x$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$\ln (x) \frac{x^{3}}{3} - \int \frac{1}{3} x^{3} \frac{1}{x} d x$

Step 3.2

Combine $\ln (x)$ and $\frac{x^{3}}{3}$ .

$\frac{\ln (x) x^{3}}{3} - \int \frac{1}{3} x^{3} \frac{1}{x} d x$

Step 4

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int x^{3} \frac{1}{x} d x)$

Step 5

Simplify.

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Step 5.1

Combine $x^{3}$ and $\frac{1}{x}$ .

$\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int \frac{x^{3}}{x} d x)$

Step 5.2

Cancel the common factor of $x^{3}$ and $x$ .

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Step 5.2.1

Factor $x$ out of $x^{3}$ .

$\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int \frac{x \cdot x^{2}}{x} d x)$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Raise $x$ to the power of $1$ .

$\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int \frac{x \cdot x^{2}}{x^{1}} d x)$

Step 5.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int \frac{x \cdot x^{2}}{x \cdot 1} d x)$

Step 5.2.2.3

Cancel the common factor.

$\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int \frac{x \cdot x^{2}}{x \cdot 1} d x)$

Step 5.2.2.4

Rewrite the expression.

$\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int \frac{x^{2}}{1} d x)$

Step 5.2.2.5

Divide $x^{2}$ by $1$ .

$\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int x^{2} d x)$

$\frac{\ln (x) x^{3}}{3} - \frac{1}{3} \int x^{2} d x$

Step 6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{1}{3} x^{3} + C)$

Step 7

Simplify the answer.

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Step 7.1

Rewrite $\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{1}{3} x^{3} + C)$ as $\frac{1}{3} \ln (x) x^{3} - \frac{1}{3} \cdot \frac{1}{3} x^{3} + C$ .

$\frac{1}{3} \ln (x) x^{3} - \frac{1}{3} \cdot \frac{1}{3} x^{3} + C$

Step 7.2

Simplify.

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Step 7.2.1

Combine $\frac{1}{3}$ and $\ln (x)$ .

$\frac{\ln (x)}{3} x^{3} - \frac{1}{3} \cdot \frac{1}{3} x^{3} + C$

Step 7.2.2

Combine $\frac{\ln (x)}{3}$ and $x^{3}$ .

$\frac{\ln (x) x^{3}}{3} - \frac{1}{3} \cdot \frac{1}{3} x^{3} + C$

Step 7.2.3

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$\frac{\ln (x) x^{3}}{3} - \frac{1}{3 \cdot 3} x^{3} + C$

Step 7.2.4

Multiply $3$ by $3$ .

$\frac{\ln (x) x^{3}}{3} - \frac{1}{9} x^{3} + C$

Step 7.3

Combine $x^{3}$ and $\frac{1}{9}$ .

$\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9} + C$

Step 7.4

Reorder terms.

$\frac{1}{3} \ln (x) x^{3} - \frac{1}{9} x^{3} + C$