Calculus Examples

$\frac{d y}{d x} = {(e^{x} - e^{- x})}^{2}$

Step 1

Rewrite the equation.

$d y = {(e^{x} - e^{- x})}^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

$y + C_{1} = \int {(e^{x} - e^{- x})}^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify.

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Apply the distributive property.

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

Step 2.3.1.2.3

Apply the distributive property.

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

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

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

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Step 2.3.1.3.1.1.1

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

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

Step 2.3.1.3.1.1.2

Add $x$ and $x$ .

$y + C_{1} = \int e^{2 x} + e^{x} (- e^{- x}) - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.2

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int e^{2 x} - e^{x} e^{- x} - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.3

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

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Step 2.3.1.3.1.3.1

Move $e^{- x}$ .

$y + C_{1} = \int e^{2 x} - (e^{- x} e^{x}) - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.3.2

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

$y + C_{1} = \int e^{2 x} - e^{- x + x} - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.3.3

Add $- x$ and $x$ .

$y + C_{1} = \int e^{2 x} - e^{0} - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.4

Simplify $- e^{0}$ .

$y + C_{1} = \int e^{2 x} - 1 - e^{- x} e^{x} - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.5

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

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Step 2.3.1.3.1.5.1

Move $e^{x}$ .

$y + C_{1} = \int e^{2 x} - 1 - (e^{x} e^{- x}) - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.5.2

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

$y + C_{1} = \int e^{2 x} - 1 - e^{x - x} - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.5.3

Subtract $x$ from $x$ .

$y + C_{1} = \int e^{2 x} - 1 - e^{0} - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.6

Simplify $- e^{0}$ .

$y + C_{1} = \int e^{2 x} - 1 - 1 - e^{- x} (- e^{- x}) d x$

Step 2.3.1.3.1.7

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int e^{2 x} - 1 - 1 - 1 \cdot - 1 e^{- x} e^{- x} d x$

Step 2.3.1.3.1.8

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

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Step 2.3.1.3.1.8.1

Move $e^{- x}$ .

$y + C_{1} = \int e^{2 x} - 1 - 1 - 1 \cdot - 1 (e^{- x} e^{- x}) d x$

Step 2.3.1.3.1.8.2

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

$y + C_{1} = \int e^{2 x} - 1 - 1 - 1 \cdot - 1 e^{- x - x} d x$

Step 2.3.1.3.1.8.3

Subtract $x$ from $- x$ .

$y + C_{1} = \int e^{2 x} - 1 - 1 - 1 \cdot - 1 e^{- 2 x} d x$

Step 2.3.1.3.1.9

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.3.1.10

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

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

Step 2.3.1.3.2

Subtract $1$ from $- 1$ .

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

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 2.3.3.1

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

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Step 2.3.3.1.1

Differentiate $2 x$ .

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

Step 2.3.3.1.2

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 2.3.3.1.3

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

$2 \cdot 1$

Step 2.3.3.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.3.2

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

$y + C_{1} = \int e^{u_{1}} \frac{1}{2} d u_{1} + \int - 2 d x + \int e^{- 2 x} d x$

Step 2.3.4

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

$y + C_{1} = \int \frac{e^{u_{1}}}{2} d u_{1} + \int - 2 d x + \int e^{- 2 x} d x$

Step 2.3.5

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

$y + C_{1} = \frac{1}{2} \int e^{u_{1}} d u_{1} + \int - 2 d x + \int e^{- 2 x} d x$

Step 2.3.6

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

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

Step 2.3.7

Apply the constant rule.

$y + C_{1} = \frac{1}{2} (e^{u_{1}} + C_{2}) - 2 x + C_{3} + \int e^{- 2 x} d x$

Step 2.3.8

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

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Step 2.3.8.1

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

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Step 2.3.8.1.1

Differentiate $- 2 x$ .

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

Step 2.3.8.1.2

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 2.3.8.1.3

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

$- 2 \cdot 1$

Step 2.3.8.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 2.3.8.2

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

$y + C_{1} = \frac{1}{2} (e^{u_{1}} + C_{2}) - 2 x + C_{3} + \int e^{u_{2}} \frac{1}{- 2} d u_{2}$

Step 2.3.9

Simplify.

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Step 2.3.9.1

Move the negative in front of the fraction.

$y + C_{1} = \frac{1}{2} (e^{u_{1}} + C_{2}) - 2 x + C_{3} + \int e^{u_{2}} (- \frac{1}{2}) d u_{2}$

Step 2.3.9.2

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

$y + C_{1} = \frac{1}{2} (e^{u_{1}} + C_{2}) - 2 x + C_{3} + \int - \frac{e^{u_{2}}}{2} d u_{2}$

Step 2.3.10

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

$y + C_{1} = \frac{1}{2} (e^{u_{1}} + C_{2}) - 2 x + C_{3} - \int \frac{e^{u_{2}}}{2} d u_{2}$

Step 2.3.11

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

$y + C_{1} = \frac{1}{2} (e^{u_{1}} + C_{2}) - 2 x + C_{3} - (\frac{1}{2} \int e^{u_{2}} d u_{2})$

Step 2.3.12

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

$y + C_{1} = \frac{1}{2} (e^{u_{1}} + C_{2}) - 2 x + C_{3} - \frac{1}{2} (e^{u_{2}} + C_{4})$

Step 2.3.13

Simplify.

$y + C_{1} = \frac{1}{2} e^{u_{1}} - 2 x - \frac{1}{2} e^{u_{2}} + C_{5}$

Step 2.3.14

Substitute back in for each integration substitution variable.

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Step 2.3.14.1

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

$y + C_{1} = \frac{1}{2} e^{2 x} - 2 x - \frac{1}{2} e^{u_{2}} + C_{5}$

Step 2.3.14.2

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

$y + C_{1} = \frac{1}{2} e^{2 x} - 2 x - \frac{1}{2} e^{- 2 x} + C_{5}$

Step 2.3.15

Reorder terms.

$y + C_{1} = \frac{1}{2} e^{2 x} - \frac{1}{2} e^{- 2 x} - 2 x + C_{5}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = \frac{1}{2} e^{2 x} - \frac{1}{2} e^{- 2 x} - 2 x + K$