Calculus Examples

$e^{x} - e^{- x}$

Step 1

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

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $- x$ .

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

Step 7.1.2

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 7.1.3

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 7.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.2

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

$e^{x} + C - \int - e^{u} d u$

Step 8

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

$e^{x} + C - - \int e^{u} d u$

Step 9

Simplify.

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Step 9.1

Multiply $- 1$ by $- 1$ .

$e^{x} + C + 1 \int e^{u} d u$

Step 9.2

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

$e^{x} + C + \int e^{u} d u$

Step 10

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

$e^{x} + C + e^{u} + C$

Step 11

Simplify.

$e^{x} + e^{u} + C$

Step 12

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

$e^{x} + e^{- x} + C$

Step 13

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

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