Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{5}$ .

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Combine $\frac{1}{6}$ and $x^{6}$ .

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

Step 2.2

Combine $\ln (x)$ and $\frac{x^{6}}{6}$ .

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

Step 3

Since $\frac{1}{6}$ is constant with respect to $x$ , move $\frac{1}{6}$ out of the integral.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Combine $x^{6}$ and $\frac{1}{x}$ .

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

Step 4.2

Cancel the common factor of $x^{6}$ and $x$ .

Tap for more steps...

Step 4.2.1

Factor $x$ out of $x^{6}$ .

$\frac{\ln (x) x^{6}}{6} - (\frac{1}{6} \int \frac{x \cdot x^{5}}{x} d x)$

Step 4.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1

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

$\frac{\ln (x) x^{6}}{6} - (\frac{1}{6} \int \frac{x \cdot x^{5}}{x^{1}} d x)$

Step 4.2.2.2

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

$\frac{\ln (x) x^{6}}{6} - (\frac{1}{6} \int \frac{x \cdot x^{5}}{x \cdot 1} d x)$

Step 4.2.2.3

Cancel the common factor.

$\frac{\ln (x) x^{6}}{6} - (\frac{1}{6} \int \frac{x \cdot x^{5}}{x \cdot 1} d x)$

Step 4.2.2.4

Rewrite the expression.

$\frac{\ln (x) x^{6}}{6} - (\frac{1}{6} \int \frac{x^{5}}{1} d x)$

Step 4.2.2.5

Divide $x^{5}$ by $1$ .

$\frac{\ln (x) x^{6}}{6} - (\frac{1}{6} \int x^{5} d x)$

$\frac{\ln (x) x^{6}}{6} - \frac{1}{6} \int x^{5} d x$

Step 5

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$\frac{\ln (x) x^{6}}{6} - \frac{1}{6} (\frac{1}{6} x^{6} + C)$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Rewrite $\frac{\ln (x) x^{6}}{6} - \frac{1}{6} (\frac{1}{6} x^{6} + C)$ as $\frac{1}{6} \ln (x) x^{6} - \frac{1}{6} \cdot \frac{1}{6} x^{6} + C$ .

$\frac{1}{6} \ln (x) x^{6} - \frac{1}{6} \cdot \frac{1}{6} x^{6} + C$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Combine $\frac{1}{6}$ and $\ln (x)$ .

$\frac{\ln (x)}{6} x^{6} - \frac{1}{6} \cdot \frac{1}{6} x^{6} + C$

Step 6.2.2

Combine $\frac{\ln (x)}{6}$ and $x^{6}$ .

$\frac{\ln (x) x^{6}}{6} - \frac{1}{6} \cdot \frac{1}{6} x^{6} + C$

Step 6.2.3

Multiply $\frac{1}{6}$ by $\frac{1}{6}$ .

$\frac{\ln (x) x^{6}}{6} - \frac{1}{6 \cdot 6} x^{6} + C$

Step 6.2.4

Multiply $6$ by $6$ .

$\frac{\ln (x) x^{6}}{6} - \frac{1}{36} x^{6} + C$

$\frac{1}{6} \ln (x) x^{6} - \frac{1}{36} x^{6} + C$