Calculus Examples

$\int \frac{e^{r}}{1 + e^{r}} d r$

Step 1

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

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $1 + e^{r}$ .

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

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

$0 + e^{r}$

Step 1.1.4

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

$e^{r}$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u} d u$

Step 2

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

$\ln (| u |) + C$

Step 3

Replace all occurrences of $u$ with $1 + e^{r}$ .

$\ln (| 1 + e^{r} |) + C$