Calculus Examples

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

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

Expand $(e^{2 x} - e^{- 2 x}) (e^{2 x} - e^{- 2 x})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $e^{2 x}$ by $e^{2 x}$ by adding the exponents.

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

Add $2 x$ and $2 x$ .

$\int e^{4 x} + e^{2 x} (- e^{- 2 x}) - e^{- 2 x} e^{2 x} - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$\int e^{4 x} - e^{2 x} e^{- 2 x} - e^{- 2 x} e^{2 x} - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.3

Multiply $e^{2 x}$ by $e^{- 2 x}$ by adding the exponents.

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Step 1.3.1.3.1

Move $e^{- 2 x}$ .

$\int e^{4 x} - (e^{- 2 x} e^{2 x}) - e^{- 2 x} e^{2 x} - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.3.2

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

$\int e^{4 x} - e^{- 2 x + 2 x} - e^{- 2 x} e^{2 x} - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.3.3

Add $- 2 x$ and $2 x$ .

$\int e^{4 x} - e^{0} - e^{- 2 x} e^{2 x} - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.4

Simplify $- e^{0}$ .

$\int e^{4 x} - 1 - e^{- 2 x} e^{2 x} - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.5

Multiply $e^{- 2 x}$ by $e^{2 x}$ by adding the exponents.

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Step 1.3.1.5.1

Move $e^{2 x}$ .

$\int e^{4 x} - 1 - (e^{2 x} e^{- 2 x}) - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.5.2

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

$\int e^{4 x} - 1 - e^{2 x - 2 x} - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.5.3

Subtract $2 x$ from $2 x$ .

$\int e^{4 x} - 1 - e^{0} - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.6

Simplify $- e^{0}$ .

$\int e^{4 x} - 1 - 1 - e^{- 2 x} (- e^{- 2 x}) d x$

Step 1.3.1.7

Rewrite using the commutative property of multiplication.

$\int e^{4 x} - 1 - 1 - 1 \cdot - 1 e^{- 2 x} e^{- 2 x} d x$

Step 1.3.1.8

Multiply $e^{- 2 x}$ by $e^{- 2 x}$ by adding the exponents.

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Step 1.3.1.8.1

Move $e^{- 2 x}$ .

$\int e^{4 x} - 1 - 1 - 1 \cdot - 1 (e^{- 2 x} e^{- 2 x}) d x$

Step 1.3.1.8.2

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

$\int e^{4 x} - 1 - 1 - 1 \cdot - 1 e^{- 2 x - 2 x} d x$

Step 1.3.1.8.3

Subtract $2 x$ from $- 2 x$ .

$\int e^{4 x} - 1 - 1 - 1 \cdot - 1 e^{- 4 x} d x$

Step 1.3.1.9

Multiply $- 1$ by $- 1$ .

$\int e^{4 x} - 1 - 1 + 1 e^{- 4 x} d x$

Step 1.3.1.10

Multiply $e^{- 4 x}$ by $1$ .

$\int e^{4 x} - 1 - 1 + e^{- 4 x} d x$

Step 1.3.2

Subtract $1$ from $- 1$ .

$\int e^{4 x} - 2 + e^{- 4 x} d x$

Step 2

Split the single integral into multiple integrals.

$\int e^{4 x} d x + \int - 2 d x + \int e^{- 4 x} d x$

Step 3

Let $u_{1} = 4 x$ . Then $d u_{1} = 4 d x$ , so $\frac{1}{4} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.1

Let $u_{1} = 4 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 3.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 3.1.3

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

$4 \cdot 1$

Step 3.1.4

Multiply $4$ by $1$ .

$4$

Step 3.2

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

$\int e^{u_{1}} \frac{1}{4} d u_{1} + \int - 2 d x + \int e^{- 4 x} d x$

Step 4

Combine $e^{u_{1}}$ and $\frac{1}{4}$ .

$\int \frac{e^{u_{1}}}{4} d u_{1} + \int - 2 d x + \int e^{- 4 x} d x$

Step 5

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int e^{u_{1}} d u_{1} + \int - 2 d x + \int e^{- 4 x} d x$

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Let $u_{2} = - 4 x$ . Then $d u_{2} = - 4 d x$ , so $- \frac{1}{4} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 8.1

Let $u_{2} = - 4 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 8.1.1

Differentiate $- 4 x$ .

$\frac{d}{d x} [- 4 x]$

Step 8.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 4 \frac{d}{d x} [x]$

Step 8.1.3

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

$- 4 \cdot 1$

Step 8.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 8.2

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

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

Step 9

Simplify.

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Step 9.1

Move the negative in front of the fraction.

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

Step 9.2

Combine $e^{u_{2}}$ and $\frac{1}{4}$ .

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

Step 10

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

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

Step 11

Since $\frac{1}{4}$ is constant with respect to $u_{2}$ , move $\frac{1}{4}$ out of the integral.

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

Step 12

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

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

Step 13

Simplify.

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

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u_{1}$ with $4 x$ .

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

Step 14.2

Replace all occurrences of $u_{2}$ with $- 4 x$ .

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