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

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

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

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $- x$ .

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

Step 5.1.2

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 5.1.3

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 5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 5.2

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Multiply $- 1$ by $- 1$ .

$1 \int e^{u} d u$

Step 7.2

Multiply $\int e^{u} d u$ by $1$ .

$\int e^{u} d u$

Step 8

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

$e^{u} + C$

Step 9

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

$e^{- x} + C$

Step 10

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

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