Calculus Examples

$\int x^{6} \ln (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{6}$ .

$\ln (x) (\frac{1}{7} x^{7}) - \int \frac{1}{7} x^{7} \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{7}$ and $x^{7}$ .

$\ln (x) \frac{x^{7}}{7} - \int \frac{1}{7} x^{7} \frac{1}{x} d x$

Step 2.2

Combine $\ln (x)$ and $\frac{x^{7}}{7}$ .

$\frac{\ln (x) x^{7}}{7} - \int \frac{1}{7} x^{7} \frac{1}{x} d x$

Step 3

Since $\frac{1}{7}$ is constant with respect to $x$ , move $\frac{1}{7}$ out of the integral.

$\frac{\ln (x) x^{7}}{7} - (\frac{1}{7} \int x^{7} \frac{1}{x} d x)$

Step 4

Simplify.

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Step 4.1

Combine $x^{7}$ and $\frac{1}{x}$ .

$\frac{\ln (x) x^{7}}{7} - (\frac{1}{7} \int \frac{x^{7}}{x} d x)$

Step 4.2

Cancel the common factor of $x^{7}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{7}$ .

$\frac{\ln (x) x^{7}}{7} - (\frac{1}{7} \int \frac{x \cdot x^{6}}{x} d x)$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

$\frac{\ln (x) x^{7}}{7} - (\frac{1}{7} \int \frac{x \cdot x^{6}}{x^{1}} d x)$

Step 4.2.2.2

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

$\frac{\ln (x) x^{7}}{7} - (\frac{1}{7} \int \frac{x \cdot x^{6}}{x \cdot 1} d x)$

Step 4.2.2.3

Cancel the common factor.

$\frac{\ln (x) x^{7}}{7} - (\frac{1}{7} \int \frac{x \cdot x^{6}}{x \cdot 1} d x)$

Step 4.2.2.4

Rewrite the expression.

$\frac{\ln (x) x^{7}}{7} - (\frac{1}{7} \int \frac{x^{6}}{1} d x)$

Step 4.2.2.5

Divide $x^{6}$ by $1$ .

$\frac{\ln (x) x^{7}}{7} - (\frac{1}{7} \int x^{6} d x)$

$\frac{\ln (x) x^{7}}{7} - \frac{1}{7} \int x^{6} d x$

Step 5

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$\frac{\ln (x) x^{7}}{7} - \frac{1}{7} (\frac{1}{7} x^{7} + C)$

Step 6

Simplify the answer.

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Step 6.1

Rewrite $\frac{\ln (x) x^{7}}{7} - \frac{1}{7} (\frac{1}{7} x^{7} + C)$ as $\frac{1}{7} \ln (x) x^{7} - \frac{1}{7} \cdot \frac{1}{7} x^{7} + C$ .

$\frac{1}{7} \ln (x) x^{7} - \frac{1}{7} \cdot \frac{1}{7} x^{7} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $\frac{1}{7}$ and $\ln (x)$ .

$\frac{\ln (x)}{7} x^{7} - \frac{1}{7} \cdot \frac{1}{7} x^{7} + C$

Step 6.2.2

Combine $\frac{\ln (x)}{7}$ and $x^{7}$ .

$\frac{\ln (x) x^{7}}{7} - \frac{1}{7} \cdot \frac{1}{7} x^{7} + C$

Step 6.2.3

Multiply $\frac{1}{7}$ by $\frac{1}{7}$ .

$\frac{\ln (x) x^{7}}{7} - \frac{1}{7 \cdot 7} x^{7} + C$

Step 6.2.4

Multiply $7$ by $7$ .

$\frac{\ln (x) x^{7}}{7} - \frac{1}{49} x^{7} + C$

$\frac{1}{7} \ln (x) x^{7} - \frac{1}{49} x^{7} + C$