Calculus Examples

$\frac{d y}{d x} = {(7 e^{x} - 5 e^{- x})}^{2}$

Step 1

Rewrite the equation.

$d y = {(7 e^{x} - 5 e^{- x})}^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int {(7 e^{x} - 5 e^{- x})}^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int {(7 e^{x} - 5 e^{- x})}^{2} d x$

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 ${(7 e^{x} - 5 e^{- x})}^{2}$ as $(7 e^{x} - 5 e^{- x}) (7 e^{x} - 5 e^{- x})$ .

$y + C_{1} = \int (7 e^{x} - 5 e^{- x}) (7 e^{x} - 5 e^{- x}) d x$

Step 2.3.1.2

Expand $(7 e^{x} - 5 e^{- x}) (7 e^{x} - 5 e^{- x})$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$y + C_{1} = \int 7 e^{x} (7 e^{x} - 5 e^{- x}) - 5 e^{- x} (7 e^{x} - 5 e^{- x}) d x$

Step 2.3.1.2.2

Apply the distributive property.

$y + C_{1} = \int 7 e^{x} (7 e^{x}) + 7 e^{x} (- 5 e^{- x}) - 5 e^{- x} (7 e^{x} - 5 e^{- x}) d x$

Step 2.3.1.2.3

Apply the distributive property.

$y + C_{1} = \int 7 e^{x} (7 e^{x}) + 7 e^{x} (- 5 e^{- x}) - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

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

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int 7 \cdot 7 e^{x} e^{x} + 7 e^{x} (- 5 e^{- x}) - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.2

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

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Step 2.3.1.3.1.2.1

Move $e^{x}$ .

$y + C_{1} = \int 7 \cdot 7 (e^{x} e^{x}) + 7 e^{x} (- 5 e^{- x}) - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.2.2

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

$y + C_{1} = \int 7 \cdot 7 e^{x + x} + 7 e^{x} (- 5 e^{- x}) - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.2.3

Add $x$ and $x$ .

$y + C_{1} = \int 7 \cdot 7 e^{2 x} + 7 e^{x} (- 5 e^{- x}) - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.3

Multiply $7$ by $7$ .

$y + C_{1} = \int 49 e^{2 x} + 7 e^{x} (- 5 e^{- x}) - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 5 e^{x} e^{- x} - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

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 $e^{- x}$ .

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 5 (e^{- x} e^{x}) - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.5.2

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

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 5 e^{- x + x} - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.5.3

Add $- x$ and $x$ .

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 5 e^{0} - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.6

Simplify $7 \cdot - 5 e^{0}$ .

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 5 - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.7

Multiply $7$ by $- 5$ .

$y + C_{1} = \int 49 e^{2 x} - 35 - 5 e^{- x} (7 e^{x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.8

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int 49 e^{2 x} - 35 - 5 \cdot 7 e^{- x} e^{x} - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.9

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

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Step 2.3.1.3.1.9.1

Move $e^{x}$ .

$y + C_{1} = \int 49 e^{2 x} - 35 - 5 \cdot 7 (e^{x} e^{- x}) - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.9.2

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

$y + C_{1} = \int 49 e^{2 x} - 35 - 5 \cdot 7 e^{x - x} - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.9.3

Subtract $x$ from $x$ .

$y + C_{1} = \int 49 e^{2 x} - 35 - 5 \cdot 7 e^{0} - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.10

Simplify $- 5 \cdot 7 e^{0}$ .

$y + C_{1} = \int 49 e^{2 x} - 35 - 5 \cdot 7 - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.11

Multiply $- 5$ by $7$ .

$y + C_{1} = \int 49 e^{2 x} - 35 - 35 - 5 e^{- x} (- 5 e^{- x}) d x$

Step 2.3.1.3.1.12

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int 49 e^{2 x} - 35 - 35 - 5 \cdot - 5 e^{- x} e^{- x} d x$

Step 2.3.1.3.1.13

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

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Step 2.3.1.3.1.13.1

Move $e^{- x}$ .

$y + C_{1} = \int 49 e^{2 x} - 35 - 35 - 5 \cdot - 5 (e^{- x} e^{- x}) d x$

Step 2.3.1.3.1.13.2

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

$y + C_{1} = \int 49 e^{2 x} - 35 - 35 - 5 \cdot - 5 e^{- x - x} d x$

Step 2.3.1.3.1.13.3

Subtract $x$ from $- x$ .

$y + C_{1} = \int 49 e^{2 x} - 35 - 35 - 5 \cdot - 5 e^{- 2 x} d x$

Step 2.3.1.3.1.14

Multiply $- 5$ by $- 5$ .

$y + C_{1} = \int 49 e^{2 x} - 35 - 35 + 25 e^{- 2 x} d x$

Step 2.3.1.3.2

Subtract $35$ from $- 35$ .

$y + C_{1} = \int 49 e^{2 x} - 70 + 25 e^{- 2 x} d x$

Step 2.3.2

Split the single integral into multiple integrals.

$y + C_{1} = \int 49 e^{2 x} d x + \int - 70 d x + \int 25 e^{- 2 x} d x$

Step 2.3.3

Since $49$ is constant with respect to $x$ , move $49$ out of the integral.

$y + C_{1} = 49 \int e^{2 x} d x + \int - 70 d x + \int 25 e^{- 2 x} d x$

Step 2.3.4

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 2.3.4.1

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

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Step 2.3.4.1.1

Differentiate $2 x$ .

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

Step 2.3.4.1.2

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 2.3.4.1.3

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 2.3.4.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.4.2

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

$y + C_{1} = 49 \int e^{u_{1}} \frac{1}{2} d u_{1} + \int - 70 d x + \int 25 e^{- 2 x} d x$

Step 2.3.5

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

$y + C_{1} = 49 \int \frac{e^{u_{1}}}{2} d u_{1} + \int - 70 d x + \int 25 e^{- 2 x} d x$

Step 2.3.6

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

$y + C_{1} = 49 (\frac{1}{2} \int e^{u_{1}} d u_{1}) + \int - 70 d x + \int 25 e^{- 2 x} d x$

Step 2.3.7

Combine $\frac{1}{2}$ and $49$ .

$y + C_{1} = \frac{49}{2} \int e^{u_{1}} d u_{1} + \int - 70 d x + \int 25 e^{- 2 x} d x$

Step 2.3.8

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

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) + \int - 70 d x + \int 25 e^{- 2 x} d x$

Step 2.3.9

Apply the constant rule.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} + \int 25 e^{- 2 x} d x$

Step 2.3.10

Since $25$ is constant with respect to $x$ , move $25$ out of the integral.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} + 25 \int e^{- 2 x} d x$

Step 2.3.11

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 2.3.11.1

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

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Step 2.3.11.1.1

Differentiate $- 2 x$ .

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

Step 2.3.11.1.2

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 2.3.11.1.3

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 2.3.11.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 2.3.11.2

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

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} + 25 \int e^{u_{2}} \frac{1}{- 2} d u_{2}$

Step 2.3.12

Simplify.

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Step 2.3.12.1

Move the negative in front of the fraction.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} + 25 \int e^{u_{2}} (- \frac{1}{2}) d u_{2}$

Step 2.3.12.2

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

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} + 25 \int - \frac{e^{u_{2}}}{2} d u_{2}$

Step 2.3.13

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

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} + 25 (- \int \frac{e^{u_{2}}}{2} d u_{2})$

Step 2.3.14

Multiply $- 1$ by $25$ .

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} - 25 \int \frac{e^{u_{2}}}{2} d u_{2}$

Step 2.3.15

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

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} - 25 (\frac{1}{2} \int e^{u_{2}} d u_{2})$

Step 2.3.16

Simplify.

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Step 2.3.16.1

Combine $\frac{1}{2}$ and $- 25$ .

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} + \frac{- 25}{2} \int e^{u_{2}} d u_{2}$

Step 2.3.16.2

Move the negative in front of the fraction.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} - \frac{25}{2} \int e^{u_{2}} d u_{2}$

Step 2.3.17

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

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 70 x + C_{3} - \frac{25}{2} (e^{u_{2}} + C_{4})$

Step 2.3.18

Simplify.

$y + C_{1} = \frac{49}{2} e^{u_{1}} - 70 x - \frac{25}{2} e^{u_{2}} + C_{5}$

Step 2.3.19

Substitute back in for each integration substitution variable.

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Step 2.3.19.1

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

$y + C_{1} = \frac{49}{2} e^{2 x} - 70 x - \frac{25}{2} e^{u_{2}} + C_{5}$

Step 2.3.19.2

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

$y + C_{1} = \frac{49}{2} e^{2 x} - 70 x - \frac{25}{2} e^{- 2 x} + C_{5}$

Step 2.3.20

Reorder terms.

$y + C_{1} = \frac{49}{2} e^{2 x} - \frac{25}{2} e^{- 2 x} - 70 x + C_{5}$

Step 2.4

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

$y = \frac{49}{2} e^{2 x} - \frac{25}{2} e^{- 2 x} - 70 x + K$