Calculus Examples

$\int {(e^{x} - e^{- x})}^{2} d x$

Step 1

Rewrite ${(e^{x} - e^{- x})}^{2}$ as $(e^{x} - e^{- x}) (e^{x} - e^{- x})$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

Add $x$ and $x$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

$\int e^{2 x} - e^{x} e^{- x} - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 3.1.3

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

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Step 3.1.3.1

Move $e^{- x}$ .

$\int e^{2 x} - (e^{- x} e^{x}) - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 3.1.3.2

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

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

Step 3.1.3.3

Add $- x$ and $x$ .

$\int e^{2 x} - e^{0} - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 3.1.4

Simplify $- e^{0}$ .

$\int e^{2 x} - 1 - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 3.1.5

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

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Step 3.1.5.1

Move $e^{x}$ .

$\int e^{2 x} - 1 - (e^{x} e^{- x}) - e^{- x} (- e^{- x}) d x$

Step 3.1.5.2

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

$\int e^{2 x} - 1 - e^{x - x} - e^{- x} (- e^{- x}) d x$

Step 3.1.5.3

Subtract $x$ from $x$ .

$\int e^{2 x} - 1 - e^{0} - e^{- x} (- e^{- x}) d x$

Step 3.1.6

Simplify $- e^{0}$ .

$\int e^{2 x} - 1 - 1 - e^{- x} (- e^{- x}) d x$

Step 3.1.7

Rewrite using the commutative property of multiplication.

$\int e^{2 x} - 1 - 1 - 1 \cdot - 1 e^{- x} e^{- x} d x$

Step 3.1.8

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

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Step 3.1.8.1

Move $e^{- x}$ .

$\int e^{2 x} - 1 - 1 - 1 \cdot - 1 (e^{- x} e^{- x}) d x$

Step 3.1.8.2

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

$\int e^{2 x} - 1 - 1 - 1 \cdot - 1 e^{- x - x} d x$

Step 3.1.8.3

Subtract $x$ from $- x$ .

$\int e^{2 x} - 1 - 1 - 1 \cdot - 1 e^{- 2 x} d x$

Step 3.1.9

Multiply $- 1$ by $- 1$ .

$\int e^{2 x} - 1 - 1 + 1 e^{- 2 x} d x$

Step 3.1.10

Multiply $e^{- 2 x}$ by $1$ .

$\int e^{2 x} - 1 - 1 + e^{- 2 x} d x$

Step 3.2

Subtract $1$ from $- 1$ .

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

Step 4

Let $u_{1} = 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 4.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 4.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 4.1.3

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

$2 \cdot 1$

Step 4.1.4

Multiply $2$ by $1$ .

$2$

Step 4.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int (e^{u_{1}} - 2 + e^{- 2 \frac{u_{1}}{2}}) \frac{1}{2} d u_{1}$

Step 5

Simplify.

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Step 5.1

Combine $- 2$ and $\frac{u_{1}}{2}$ .

$\int (e^{u_{1}} - 2 + e^{\frac{- 2 u_{1}}{2}}) \frac{1}{2} d u_{1}$

Step 5.2

Cancel the common factor of $- 2$ and $2$ .

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Step 5.2.1

Factor $2$ out of $- 2 u_{1}$ .

$\int (e^{u_{1}} - 2 + e^{\frac{2 (- u_{1})}{2}}) \frac{1}{2} d u_{1}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

$\int (e^{u_{1}} - 2 + e^{\frac{2 (- u_{1})}{2 (1)}}) \frac{1}{2} d u_{1}$

Step 5.2.2.2

Cancel the common factor.

$\int (e^{u_{1}} - 2 + e^{\frac{2 (- u_{1})}{2 \cdot 1}}) \frac{1}{2} d u_{1}$

Step 5.2.2.3

Rewrite the expression.

$\int (e^{u_{1}} - 2 + e^{\frac{- u_{1}}{1}}) \frac{1}{2} d u_{1}$

Step 5.2.2.4

Divide $- u_{1}$ by $1$ .

$\int (e^{u_{1}} - 2 + e^{- u_{1}}) \frac{1}{2} d u_{1}$

Step 6

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int e^{u_{1}} - 2 + e^{- u_{1}} d u_{1}$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{2} (\int e^{u_{1}} d u_{1} + \int - 2 d u_{1} + \int e^{- u_{1}} d u_{1})$

Step 8

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$\frac{1}{2} (e^{u_{1}} + C + \int - 2 d u_{1} + \int e^{- u_{1}} d u_{1})$

Step 9

Apply the constant rule.

$\frac{1}{2} (e^{u_{1}} + C - 2 u_{1} + C + \int e^{- u_{1}} d u_{1})$

Step 10

Let $u_{2} = - u_{1}$ . Then $d u_{2} = - d u_{1}$ , so $- d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 10.1

Let $u_{2} = - u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 10.1.1

Differentiate $- u_{1}$ .

$\frac{d}{d u_{1}} [- u_{1}]$

Step 10.1.2

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

$- \frac{d}{d u_{1}} [u_{1}]$

Step 10.1.3

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

$- 1 \cdot 1$

Step 10.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 10.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} (e^{u_{1}} + C - 2 u_{1} + C + \int - e^{u_{2}} d u_{2})$

Step 11

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$\frac{1}{2} (e^{u_{1}} + C - 2 u_{1} + C - \int e^{u_{2}} d u_{2})$

Step 12

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$\frac{1}{2} (e^{u_{1}} + C - 2 u_{1} + C - (e^{u_{2}} + C))$

Step 13

Simplify.

$\frac{1}{2} (e^{u_{1}} - 2 u_{1} - e^{u_{2}}) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u_{1}$ with $2 x$ .

$\frac{1}{2} (e^{2 x} - 2 (2 x) - e^{u_{2}}) + C$

Step 14.2

Replace all occurrences of $u_{2}$ with $- u_{1}$ .

$\frac{1}{2} (e^{2 x} - 2 (2 x) - e^{- u_{1}}) + C$

Step 14.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$\frac{1}{2} (e^{2 x} - 2 (2 x) - e^{- (2 x)}) + C$

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

Multiply $2$ by $- 2$ .

$\frac{1}{2} (e^{2 x} - 4 x - e^{- (2 x)}) + C$

Step 15.1.2

Multiply $2$ by $- 1$ .

$\frac{1}{2} (e^{2 x} - 4 x - e^{- 2 x}) + C$

Step 15.2

Apply the distributive property.

$\frac{1}{2} e^{2 x} + \frac{1}{2} (- 4 x) + \frac{1}{2} (- e^{- 2 x}) + C$

Step 15.3

Simplify.

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Step 15.3.1

Combine $\frac{1}{2}$ and $e^{2 x}$ .

$\frac{e^{2 x}}{2} + \frac{1}{2} (- 4 x) + \frac{1}{2} (- e^{- 2 x}) + C$

Step 15.3.2

Cancel the common factor of $2$ .

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Step 15.3.2.1

Factor $2$ out of $- 4 x$ .

$\frac{e^{2 x}}{2} + \frac{1}{2} (2 (- 2 x)) + \frac{1}{2} (- e^{- 2 x}) + C$

Step 15.3.2.2

Cancel the common factor.

$\frac{e^{2 x}}{2} + \frac{1}{2} (2 (- 2 x)) + \frac{1}{2} (- e^{- 2 x}) + C$

Step 15.3.2.3

Rewrite the expression.

$\frac{e^{2 x}}{2} - 2 x + \frac{1}{2} (- e^{- 2 x}) + C$

Step 15.3.3

Combine $\frac{1}{2}$ and $e^{- 2 x}$ .

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

Step 16

Reorder terms.

$\frac{1}{2} e^{2 x} - 2 x - \frac{1}{2} e^{- 2 x} + C$