Calculus Examples

$\int \frac{1}{e^{x}} d x$

Step 1

Let $u = e^{x}$ . Then $d u = e^{x} d x$ , so $- d u = - e^{x} d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $e^{x}$ .

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

Step 1.1.2

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

$e^{x}$

Step 1.2

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Cancel the common factor of $1$ .

Tap for more steps...

Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

$\int 1 \cdot 1 d u$

Step 2.2

Multiply $1$ by $1$ .

$\int d u$

Step 3

Apply the constant rule.

$u + C$

Step 4

Replace all occurrences of $u$ with $e^{x}$ .

$e^{x} + C$