Calculus Examples

$e^{- x}$

Step 1

Write $e^{- x}$ as a function.

$f (x) = e^{- x}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int e^{- x} d x$

Step 4

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

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

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Step 4.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 4.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 4.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.2

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

$\int - e^{u} d u$

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int e^{u} d u$

Step 6

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$- (e^{u} + C)$

Step 7

Simplify.

$- e^{u} + C$

Step 8

Replace all occurrences of $u$ with $- x$ .

$- e^{- x} + C$

Step 9

The answer is the antiderivative of the function $f (x) = e^{- x}$ .

$F (x) =$ $- e^{- x} + C$