Calculus Examples

$\int x^{8} \ln (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{8}$ .

$\ln (x) (\frac{1}{9} x^{9}) - \int \frac{1}{9} x^{9} \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

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

$\ln (x) \frac{x^{9}}{9} - \int \frac{1}{9} x^{9} \frac{1}{x} d x$

Step 2.2

Combine $\ln (x)$ and $\frac{x^{9}}{9}$ .

$\frac{\ln (x) x^{9}}{9} - \int \frac{1}{9} x^{9} \frac{1}{x} d x$

Step 3

Since $\frac{1}{9}$ is constant with respect to $x$ , move $\frac{1}{9}$ out of the integral.

$\frac{\ln (x) x^{9}}{9} - (\frac{1}{9} \int x^{9} \frac{1}{x} d x)$

Step 4

Simplify.

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Step 4.1

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

$\frac{\ln (x) x^{9}}{9} - (\frac{1}{9} \int \frac{x^{9}}{x} d x)$

Step 4.2

Cancel the common factor of $x^{9}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{9}$ .

$\frac{\ln (x) x^{9}}{9} - (\frac{1}{9} \int \frac{x \cdot x^{8}}{x} d x)$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

$\frac{\ln (x) x^{9}}{9} - (\frac{1}{9} \int \frac{x \cdot x^{8}}{x^{1}} d x)$

Step 4.2.2.2

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

$\frac{\ln (x) x^{9}}{9} - (\frac{1}{9} \int \frac{x \cdot x^{8}}{x \cdot 1} d x)$

Step 4.2.2.3

Cancel the common factor.

$\frac{\ln (x) x^{9}}{9} - (\frac{1}{9} \int \frac{x \cdot x^{8}}{x \cdot 1} d x)$

Step 4.2.2.4

Rewrite the expression.

$\frac{\ln (x) x^{9}}{9} - (\frac{1}{9} \int \frac{x^{8}}{1} d x)$

Step 4.2.2.5

Divide $x^{8}$ by $1$ .

$\frac{\ln (x) x^{9}}{9} - (\frac{1}{9} \int x^{8} d x)$

$\frac{\ln (x) x^{9}}{9} - \frac{1}{9} \int x^{8} d x$

Step 5

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

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

Step 6

Simplify the answer.

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Step 6.1

Rewrite $\frac{\ln (x) x^{9}}{9} - \frac{1}{9} (\frac{1}{9} x^{9} + C)$ as $\frac{1}{9} \ln (x) x^{9} - \frac{1}{9} \cdot \frac{1}{9} x^{9} + C$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Combine $\frac{1}{9}$ and $\ln (x)$ .

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

Step 6.2.2

Combine $\frac{\ln (x)}{9}$ and $x^{9}$ .

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

Step 6.2.3

Multiply $\frac{1}{9}$ by $\frac{1}{9}$ .

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

Step 6.2.4

Multiply $9$ by $9$ .

$\frac{\ln (x) x^{9}}{9} - \frac{1}{81} x^{9} + C$

$\frac{1}{9} \ln (x) x^{9} - \frac{1}{81} x^{9} + C$