Calculus Examples

$\int_{1}^{4} \frac{12 \ln^{5} (x)}{x} d x$

Step 1

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int_{1}^{4} \frac{\ln^{5} (x)}{x} d x$

Step 2

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 2.1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x}$

Step 2.2

Substitute the lower limit in for $x$ in $u = \ln (x)$ .

$u_{lower} = \ln (1)$

Step 2.3

The natural logarithm of $1$ is $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = \ln (x)$ .

$u_{upper} = \ln (4)$

Step 2.5

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = \ln (4)$

Step 2.6

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$12 \int_{0}^{\ln (4)} u^{5} d u$

Step 3

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

$12 (\frac{1}{6} u^{6}]_{0}^{\ln (4)})$

Step 4

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

$12 (\frac{u^{6}}{6}]_{0}^{\ln (4)})$

Step 5

Substitute and simplify.

Tap for more steps...

Step 5.1

Evaluate $\frac{u^{6}}{6}$ at $\ln (4)$ and at $0$ .

$12 ((\frac{\ln^{6} (4)}{6}) - \frac{0^{6}}{6})$

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Raising $0$ to any positive power yields $0$ .

$12 (\frac{\ln^{6} (4)}{6} - \frac{0}{6})$

Step 5.2.2

Cancel the common factor of $0$ and $6$ .

Tap for more steps...

Step 5.2.2.1

Factor $6$ out of $0$ .

$12 (\frac{\ln^{6} (4)}{6} - \frac{6 (0)}{6})$

Step 5.2.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.2.1

Factor $6$ out of $6$ .

$12 (\frac{\ln^{6} (4)}{6} - \frac{6 \cdot 0}{6 \cdot 1})$

Step 5.2.2.2.2

Cancel the common factor.

$12 (\frac{\ln^{6} (4)}{6} - \frac{6 \cdot 0}{6 \cdot 1})$

Step 5.2.2.2.3

Rewrite the expression.

$12 (\frac{\ln^{6} (4)}{6} - \frac{0}{1})$

Step 5.2.2.2.4

Divide $0$ by $1$ .

$12 (\frac{\ln^{6} (4)}{6} - 0)$

Step 5.2.3

Multiply $- 1$ by $0$ .

$12 (\frac{\ln^{6} (4)}{6} + 0)$

Step 5.2.4

Add $\frac{\ln^{6} (4)}{6}$ and $0$ .

$12 \frac{\ln^{6} (4)}{6}$

Step 5.2.5

Combine $12$ and $\frac{\ln^{6} (4)}{6}$ .

$\frac{12 \ln^{6} (4)}{6}$

Step 5.2.6

Cancel the common factor of $12$ and $6$ .

Tap for more steps...

Step 5.2.6.1

Factor $6$ out of $12 \ln^{6} (4)$ .

$\frac{6 (2 \ln^{6} (4))}{6}$

Step 5.2.6.2

Cancel the common factors.

Tap for more steps...

Step 5.2.6.2.1

Factor $6$ out of $6$ .

$\frac{6 (2 \ln^{6} (4))}{6 (1)}$

Step 5.2.6.2.2

Cancel the common factor.

$\frac{6 (2 \ln^{6} (4))}{6 \cdot 1}$

Step 5.2.6.2.3

Rewrite the expression.

$\frac{2 \ln^{6} (4)}{1}$

Step 5.2.6.2.4

Divide $2 \ln^{6} (4)$ by $1$ .

$2 \ln^{6} (4)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 \ln^{6} (4)$

Decimal Form:

$14.19589361 \dots$