Calculus Examples

$\int_{1}^{e^{4}} \frac{\ln ({(x)}^{3})}{x} d x$

Step 1

Rewrite $\frac{\ln (x^{3})}{x}$ as $\frac{3 \ln (x)}{x}$ .

$\int_{1}^{e^{4}} \frac{3 \ln (x)}{x} d x$

Step 2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{1}^{e^{4}} \frac{\ln (x)}{x} d x$

Step 3

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 3.1

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

Tap for more steps...

Step 3.1.1

Differentiate $\ln (x)$ .

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

Step 3.1.2

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

$\frac{1}{x}$

Step 3.2

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

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

Step 3.3

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

$u_{lower} = 0$

Step 3.4

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

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

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Use logarithm rules to move $4$ out of the exponent.

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

Step 3.5.2

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

$u_{upper} = 4 \cdot 1$

Step 3.5.3

Multiply $4$ by $1$ .

$u_{upper} = 4$

Step 3.6

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

$u_{lower} = 0$

$u_{upper} = 4$

Step 3.7

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

$3 \int_{0}^{4} u d u$

Step 4

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$3 (\frac{1}{2} u^{2}]_{0}^{4})$

Step 5

Combine $\frac{1}{2}$ and $u^{2}$ .

$3 (\frac{u^{2}}{2}]_{0}^{4})$

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

Evaluate $\frac{u^{2}}{2}$ at $4$ and at $0$ .

$3 ((\frac{4^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Raise $4$ to the power of $2$ .

$3 (\frac{16}{2} - \frac{0^{2}}{2})$

Step 6.2.2

Cancel the common factor of $16$ and $2$ .

Tap for more steps...

Step 6.2.2.1

Factor $2$ out of $16$ .

$3 (\frac{2 \cdot 8}{2} - \frac{0^{2}}{2})$

Step 6.2.2.2

Cancel the common factors.

Tap for more steps...

Step 6.2.2.2.1

Factor $2$ out of $2$ .

$3 (\frac{2 \cdot 8}{2 (1)} - \frac{0^{2}}{2})$

Step 6.2.2.2.2

Cancel the common factor.

$3 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 6.2.2.2.3

Rewrite the expression.

$3 (\frac{8}{1} - \frac{0^{2}}{2})$

Step 6.2.2.2.4

Divide $8$ by $1$ .

$3 (8 - \frac{0^{2}}{2})$

Step 6.2.3

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

$3 (8 - \frac{0}{2})$

Step 6.2.4

Cancel the common factor of $0$ and $2$ .

Tap for more steps...

Step 6.2.4.1

Factor $2$ out of $0$ .

$3 (8 - \frac{2 (0)}{2})$

Step 6.2.4.2

Cancel the common factors.

Tap for more steps...

Step 6.2.4.2.1

Factor $2$ out of $2$ .

$3 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 6.2.4.2.2

Cancel the common factor.

$3 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 6.2.4.2.3

Rewrite the expression.

$3 (8 - \frac{0}{1})$

Step 6.2.4.2.4

Divide $0$ by $1$ .

$3 (8 - 0)$

Step 6.2.5

Multiply $- 1$ by $0$ .

$3 (8 + 0)$

Step 6.2.6

Add $8$ and $0$ .

$3 \cdot 8$

Step 6.2.7

Multiply $3$ by $8$ .

$24$