Calculus Examples

$e^{x} (1 + e^{- x})$

Step 1

Write $e^{x} (1 + e^{- x})$ as a function.

$f (x) = e^{x} (1 + e^{- x})$

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.

$F (x) = \int e^{x} (1 + e^{- x}) d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\int e^{x} \cdot 1 + e^{x} e^{- x} d x$

Step 4.2

Multiply $e^{x}$ by $1$ .

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

Step 4.3

Multiply $e^{x}$ by $e^{- x}$ by adding the exponents.

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Step 4.3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.3.2

Subtract $x$ from $x$ .

$\int e^{x} + e^{0} d x$

Step 4.4

Simplify $e^{0}$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Simplify.

$e^{x} + x + C$

Step 9

The answer is the antiderivative of the function $f (x) = e^{x} (1 + e^{- x})$ .

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