Calculus Examples

$\int_{1}^{3} \frac{12 \ln^{3} (x)}{x} d x$

Step 1

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

$12 \int_{1}^{3} \frac{\ln^{3} (x)}{x} d x$

Step 2

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

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Step 2.1

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

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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 $u = \ln (x)$ .

$u_{upper} = \ln (3)$

Step 2.5

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

$u_{lower} = 0$

$u_{upper} = \ln (3)$

Step 2.6

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

$12 \int_{0}^{\ln (3)} u^{3} d u$

Step 3

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$12 (\frac{1}{4} u^{4}]_{0}^{\ln (3)})$

Step 4

Combine $\frac{1}{4}$ and $u^{4}$ .

$12 (\frac{u^{4}}{4}]_{0}^{\ln (3)})$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{u^{4}}{4}$ at $\ln (3)$ and at $0$ .

$12 ((\frac{\ln^{4} (3)}{4}) - \frac{0^{4}}{4})$

Step 5.2

Simplify.

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Step 5.2.1

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

$12 (\frac{\ln^{4} (3)}{4} - \frac{0}{4})$

Step 5.2.2

Cancel the common factor of $0$ and $4$ .

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Step 5.2.2.1

Factor $4$ out of $0$ .

$12 (\frac{\ln^{4} (3)}{4} - \frac{4 (0)}{4})$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $4$ out of $4$ .

$12 (\frac{\ln^{4} (3)}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 5.2.2.2.2

Cancel the common factor.

$12 (\frac{\ln^{4} (3)}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 5.2.2.2.3

Rewrite the expression.

$12 (\frac{\ln^{4} (3)}{4} - \frac{0}{1})$

Step 5.2.2.2.4

Divide $0$ by $1$ .

$12 (\frac{\ln^{4} (3)}{4} - 0)$

Step 5.2.3

Multiply $- 1$ by $0$ .

$12 (\frac{\ln^{4} (3)}{4} + 0)$

Step 5.2.4

Add $\frac{\ln^{4} (3)}{4}$ and $0$ .

$12 \frac{\ln^{4} (3)}{4}$

Step 5.2.5

Combine $12$ and $\frac{\ln^{4} (3)}{4}$ .

$\frac{12 \ln^{4} (3)}{4}$

Step 5.2.6

Cancel the common factor of $12$ and $4$ .

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Step 5.2.6.1

Factor $4$ out of $12 \ln^{4} (3)$ .

$\frac{4 (3 \ln^{4} (3))}{4}$

Step 5.2.6.2

Cancel the common factors.

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Step 5.2.6.2.1

Factor $4$ out of $4$ .

$\frac{4 (3 \ln^{4} (3))}{4 (1)}$

Step 5.2.6.2.2

Cancel the common factor.

$\frac{4 (3 \ln^{4} (3))}{4 \cdot 1}$

Step 5.2.6.2.3

Rewrite the expression.

$\frac{3 \ln^{4} (3)}{1}$

Step 5.2.6.2.4

Divide $3 \ln^{4} (3)$ by $1$ .

$3 \ln^{4} (3)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$3 \ln^{4} (3)$

Decimal Form:

$4.37017738 \dots$