Calculus Examples

$\int_{\ln (2)}^{\ln (8)} \frac{e^{x}}{1 + e^{x}} d x$

Step 1

Let $u = 1 + e^{x}$ . Then $d u = e^{x} d x$ , so $\frac{1}{e^{x}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 1 + e^{x}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $1 + e^{x}$ .

$\frac{d}{d x} [1 + e^{x}]$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $1 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [e^{x}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [e^{x}]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [e^{x}]$

Step 1.1.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$0 + e^{x}$

Step 1.1.4

Add $0$ and $e^{x}$ .

$e^{x}$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 + e^{x}$ .

$u_{lower} = 1 + e^{\ln (2)}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Exponentiation and log are inverse functions.

$u_{lower} = 1 + 2$

Step 1.3.2

Add $1$ and $2$ .

$u_{lower} = 3$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 + e^{x}$ .

$u_{upper} = 1 + e^{\ln (8)}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Exponentiation and log are inverse functions.

$u_{upper} = 1 + 8$

Step 1.5.2

Add $1$ and $8$ .

$u_{upper} = 9$

Step 1.6

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

$u_{lower} = 3$

$u_{upper} = 9$

Step 1.7

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

$\int_{3}^{9} \frac{1}{u} d u$

Step 2

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\ln (| u |)]_{3}^{9}$

Step 3

Evaluate $\ln (| u |)$ at $9$ and at $3$ .

$(\ln (| 9 |)) - \ln (| 3 |)$

Step 4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| 9 |}{| 3 |})$

Step 5

Simplify.

Tap for more steps...

Step 5.1

The absolute value is the distance between a number and zero. The distance between $0$ and $9$ is $9$ .

$\ln (\frac{9}{| 3 |})$

Step 5.2

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\ln (\frac{9}{3})$

Step 5.3

Divide $9$ by $3$ .

$\ln (3)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (3)$

Decimal Form:

$1.09861228 \dots$