Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Expand $(e^{x} + 1) (e^{- x} + 1)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

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Step 1.2.1.1.1

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

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

Step 1.2.1.1.2

Subtract $x$ from $x$ .

$\int e^{0} + e^{x} \cdot 1 + 1 e^{- x} + 1 \cdot 1 d x$

Step 1.2.1.2

Simplify $e^{0}$ .

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

Step 1.2.1.3

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

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

Step 1.2.1.4

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

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

Step 1.2.1.5

Multiply $1$ by $1$ .

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

Step 1.2.2

Add $1$ and $1$ .

$\int 2 + e^{x} + e^{- x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 d x + \int e^{x} d x + \int e^{- x} d x$

Step 3

Apply the constant rule.

$2 x + C + \int e^{x} d x + \int e^{- x} d x$

Step 4

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

$2 x + C + e^{x} + C + \int e^{- x} d x$

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 $d u$ .

$2 x + C + e^{x} + C + \int - e^{u} d u$

Step 6

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

$2 x + C + e^{x} + C - \int e^{u} d u$

Step 7

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

$2 x + C + e^{x} + C - (e^{u} + C)$

Step 8

Simplify.

$2 x + e^{x} - e^{u} + C$

Step 9

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

$2 x + e^{x} - e^{- x} + C$