Calculus Examples

\int x^{4} ln (x) d x

$\int x^{4} \ln (x) d x$

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^{4}

$d v = x^{4}$ .

ln (x) (\frac{1}{5} x^{5}) - \int \frac{1}{5} x^{5} \frac{1}{x} d x

$\ln (x) (\frac{1}{5} x^{5}) - \int \frac{1}{5} x^{5} \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

Combine

\frac{1}{5}

$\frac{1}{5}$ and

x^{5}

$x^{5}$ .

ln (x) \frac{x^{5}}{5} - \int \frac{1}{5} x^{5} \frac{1}{x} d x

$\ln (x) \frac{x^{5}}{5} - \int \frac{1}{5} x^{5} \frac{1}{x} d x$

Step 2.2

Combine

ln (x)

$\ln (x)$ and

\frac{x^{5}}{5}

$\frac{x^{5}}{5}$ .

\frac{ln (x) x^{5}}{5} - \int \frac{1}{5} x^{5} \frac{1}{x} d x

$\frac{\ln (x) x^{5}}{5} - \int \frac{1}{5} x^{5} \frac{1}{x} d x$

\frac{ln (x) x^{5}}{5} - \int \frac{1}{5} x^{5} \frac{1}{x} d x

$\frac{\ln (x) x^{5}}{5} - \int \frac{1}{5} x^{5} \frac{1}{x} d x$

Step 3

Since

\frac{1}{5}

$\frac{1}{5}$ is constant with respect to

x

$x$ , move

\frac{1}{5}

$\frac{1}{5}$ out of the integral.

\frac{ln (x) x^{5}}{5} - (\frac{1}{5} \int x^{5} \frac{1}{x} d x)

$\frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int x^{5} \frac{1}{x} d x)$

Step 4

Simplify.

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Step 4.1

Combine

x^{5}

$x^{5}$ and

\frac{1}{x}

$\frac{1}{x}$ .

\frac{ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x^{5}}{x} d x)

$\frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x^{5}}{x} d x)$

Step 4.2

Cancel the common factor of

x^{5}

$x^{5}$ and

x

$x$ .

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Step 4.2.1

Factor

x

$x$ out of

x^{5}

$x^{5}$ .

\frac{ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x} d x)

$\frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x} d x)$

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^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x^{1}} d x)

$\frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x^{1}} d x)$

Step 4.2.2.2

Factor

x

$x$ out of

x^{1}

$x^{1}$ .

\frac{ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x \cdot 1} d x)

$\frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x \cdot 1} d x)$

Step 4.2.2.3

Cancel the common factor.

$\frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x \cdot 1} d x)$

Step 4.2.2.4

Rewrite the expression.

$\frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x^{4}}{1} d x)$

Step 4.2.2.5

Divide

$x^{4}$ by

$1$ .

$\frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int x^{4} d x)$

$\frac{\ln (x) x^{5}}{5} - \frac{1}{5} \int x^{4} d x$

Step 5

By the Power Rule, the integral of

$x^{4}$ with respect to

$x$ is

$\frac{1}{5} x^{5}$ .

$\frac{\ln (x) x^{5}}{5} - \frac{1}{5} (\frac{1}{5} x^{5} + C)$

Step 6

Simplify the answer.

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Step 6.1

Rewrite

$\frac{\ln (x) x^{5}}{5} - \frac{1}{5} (\frac{1}{5} x^{5} + C)$ as

$\frac{1}{5} \ln (x) x^{5} - \frac{1}{5} \cdot \frac{1}{5} x^{5} + C$ .

$\frac{1}{5} \ln (x) x^{5} - \frac{1}{5} \cdot \frac{1}{5} x^{5} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine

$\frac{1}{5}$ and

$\ln (x)$ .

$\frac{\ln (x)}{5} x^{5} - \frac{1}{5} \cdot \frac{1}{5} x^{5} + C$

Step 6.2.2

Combine

$\frac{\ln (x)}{5}$ and

$x^{5}$ .

$\frac{\ln (x) x^{5}}{5} - \frac{1}{5} \cdot \frac{1}{5} x^{5} + C$

Step 6.2.3

Multiply

$\frac{1}{5}$ by

$\frac{1}{5}$ .

$\frac{\ln (x) x^{5}}{5} - \frac{1}{5 \cdot 5} x^{5} + C$

Step 6.2.4

Multiply

$5$ by

$5$ .

$\frac{\ln (x) x^{5}}{5} - \frac{1}{25} x^{5} + C$

$\frac{1}{5} \ln (x) x^{5} - \frac{1}{25} x^{5} + C$