Calculus Examples

\int x \ln (x) d x

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

Step 1

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = \ln (x)

$u = \ln (x)$ and

d v = x

$d v = x$ .

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

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

Step 2

Simplify.

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Step 2.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x^{2}

$x^{2}$ .

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

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

Step 2.2

Combine

\ln (x)

$\ln (x)$ and

\frac{x^{2}}{2}

$\frac{x^{2}}{2}$ .

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

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

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

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

Step 3

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

x

$x$ , move

\frac{1}{2}

$\frac{1}{2}$ out of the integral.

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

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

Step 4

Simplify.

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Step 4.1

Combine

x^{2}

$x^{2}$ and

\frac{1}{x}

$\frac{1}{x}$ .

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

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

Step 4.2

Cancel the common factor of

x^{2}

$x^{2}$ and

x

$x$ .

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Step 4.2.1

Factor

x

$x$ out of

x^{2}

$x^{2}$ .

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

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 4.2.2.2

Factor

x

$x$ out of

x^{1}

$x^{1}$ .

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

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

Step 4.2.2.3

Cancel the common factor.

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

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

Step 4.2.2.4

Rewrite the expression.

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

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

Step 4.2.2.5

Divide

x

$x$ by

1

$1$ .

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

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

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

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

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

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

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

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

Step 5

By the Power Rule, the integral of

x

$x$ with respect to

x

$x$ is

\frac{1}{2} x^{2}

$\frac{1}{2} x^{2}$ .

\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{2} x^{2} + C)

$\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{2} x^{2} + C)$

Step 6

Simplify the answer.

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Step 6.1

Rewrite

\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{2} x^{2} + C)

$\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{2} x^{2} + C)$ as

\frac{1}{2} \ln (x) x^{2} - \frac{1}{2} \cdot \frac{1}{2} x^{2} + C

$\frac{1}{2} \ln (x) x^{2} - \frac{1}{2} \cdot \frac{1}{2} x^{2} + C$ .

\frac{1}{2} \ln (x) x^{2} - \frac{1}{2} \cdot \frac{1}{2} x^{2} + C

$\frac{1}{2} \ln (x) x^{2} - \frac{1}{2} \cdot \frac{1}{2} x^{2} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

\ln (x)

$\ln (x)$ .

\frac{\ln (x)}{2} x^{2} - \frac{1}{2} \cdot \frac{1}{2} x^{2} + C

$\frac{\ln (x)}{2} x^{2} - \frac{1}{2} \cdot \frac{1}{2} x^{2} + C$

Step 6.2.2

Combine

\frac{\ln (x)}{2}

$\frac{\ln (x)}{2}$ and

x^{2}

$x^{2}$ .

\frac{\ln (x) x^{2}}{2} - \frac{1}{2} \cdot \frac{1}{2} x^{2} + C

$\frac{\ln (x) x^{2}}{2} - \frac{1}{2} \cdot \frac{1}{2} x^{2} + C$

Step 6.2.3

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

\frac{1}{2}

$\frac{1}{2}$ .

\frac{\ln (x) x^{2}}{2} - \frac{1}{2 \cdot 2} x^{2} + C

$\frac{\ln (x) x^{2}}{2} - \frac{1}{2 \cdot 2} x^{2} + C$

Step 6.2.4

Multiply

2

$2$ by

2

$2$ .

\frac{\ln (x) x^{2}}{2} - \frac{1}{4} x^{2} + C

$\frac{\ln (x) x^{2}}{2} - \frac{1}{4} x^{2} + C$

\frac{\ln (x) x^{2}}{2} - \frac{1}{4} x^{2} + C

$\frac{\ln (x) x^{2}}{2} - \frac{1}{4} x^{2} + C$

Step 6.3

Combine

x^{2}

$x^{2}$ and

\frac{1}{4}

$\frac{1}{4}$ .

\frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4} + C

$\frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4} + C$

Step 6.4

Reorder terms.

\frac{1}{2} \ln (x) x^{2} - \frac{1}{4} x^{2} + C

$\frac{1}{2} \ln (x) x^{2} - \frac{1}{4} x^{2} + C$

\frac{1}{2} \ln (x) x^{2} - \frac{1}{4} x^{2} + C

$\frac{1}{2} \ln (x) x^{2} - \frac{1}{4} x^{2} + C$

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