Calculus Examples

$\frac{d y}{d x} = {(7 e^{x} - 3 e^{- x})}^{2}$

Step 1

Rewrite the equation.

$d y = {(7 e^{x} - 3 e^{- x})}^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int {(7 e^{x} - 3 e^{- x})}^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int {(7 e^{x} - 3 e^{- x})}^{2} d x$

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

${(7 e^{x} - 3 e^{- x})}^{2}$ as

$(7 e^{x} - 3 e^{- x}) (7 e^{x} - 3 e^{- x})$ .

$y + C_{1} = \int (7 e^{x} - 3 e^{- x}) (7 e^{x} - 3 e^{- x}) d x$

Step 2.3.1.2

Expand

$(7 e^{x} - 3 e^{- x}) (7 e^{x} - 3 e^{- x})$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$y + C_{1} = \int 7 e^{x} (7 e^{x} - 3 e^{- x}) - 3 e^{- x} (7 e^{x} - 3 e^{- x}) d x$

Step 2.3.1.2.2

Apply the distributive property.

$y + C_{1} = \int 7 e^{x} (7 e^{x}) + 7 e^{x} (- 3 e^{- x}) - 3 e^{- x} (7 e^{x} - 3 e^{- x}) d x$

Step 2.3.1.2.3

Apply the distributive property.

$y + C_{1} = \int 7 e^{x} (7 e^{x}) + 7 e^{x} (- 3 e^{- x}) - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

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.

$y + C_{1} = \int 7 \cdot 7 e^{x} e^{x} + 7 e^{x} (- 3 e^{- x}) - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

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

$e^{x}$ .

$y + C_{1} = \int 7 \cdot 7 (e^{x} e^{x}) + 7 e^{x} (- 3 e^{- x}) - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y + C_{1} = \int 7 \cdot 7 e^{x + x} + 7 e^{x} (- 3 e^{- x}) - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.2.3

Add

$x$ and

$x$ .

$y + C_{1} = \int 7 \cdot 7 e^{2 x} + 7 e^{x} (- 3 e^{- x}) - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.3

Multiply

$7$ by

$7$ .

$y + C_{1} = \int 49 e^{2 x} + 7 e^{x} (- 3 e^{- x}) - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 3 e^{x} e^{- x} - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

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

$e^{- x}$ .

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 3 (e^{- x} e^{x}) - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.5.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 3 e^{- x + x} - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.5.3

Add

$- x$ and

$x$ .

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 3 e^{0} - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.6

Simplify

$7 \cdot - 3 e^{0}$ .

$y + C_{1} = \int 49 e^{2 x} + 7 \cdot - 3 - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.7

Multiply

$7$ by

$- 3$ .

$y + C_{1} = \int 49 e^{2 x} - 21 - 3 e^{- x} (7 e^{x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.8

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int 49 e^{2 x} - 21 - 3 \cdot 7 e^{- x} e^{x} - 3 e^{- x} (- 3 e^{- x}) d x$

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

$e^{x}$ .

$y + C_{1} = \int 49 e^{2 x} - 21 - 3 \cdot 7 (e^{x} e^{- x}) - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.9.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y + C_{1} = \int 49 e^{2 x} - 21 - 3 \cdot 7 e^{x - x} - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.9.3

Subtract

$x$ from

$x$ .

$y + C_{1} = \int 49 e^{2 x} - 21 - 3 \cdot 7 e^{0} - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.10

Simplify

$- 3 \cdot 7 e^{0}$ .

$y + C_{1} = \int 49 e^{2 x} - 21 - 3 \cdot 7 - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.11

Multiply

$- 3$ by

$7$ .

$y + C_{1} = \int 49 e^{2 x} - 21 - 21 - 3 e^{- x} (- 3 e^{- x}) d x$

Step 2.3.1.3.1.12

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int 49 e^{2 x} - 21 - 21 - 3 \cdot - 3 e^{- x} e^{- x} d x$

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

$e^{- x}$ .

$y + C_{1} = \int 49 e^{2 x} - 21 - 21 - 3 \cdot - 3 (e^{- x} e^{- x}) d x$

Step 2.3.1.3.1.13.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y + C_{1} = \int 49 e^{2 x} - 21 - 21 - 3 \cdot - 3 e^{- x - x} d x$

Step 2.3.1.3.1.13.3

Subtract

$x$ from

$- x$ .

$y + C_{1} = \int 49 e^{2 x} - 21 - 21 - 3 \cdot - 3 e^{- 2 x} d x$

Step 2.3.1.3.1.14

Multiply

$- 3$ by

$- 3$ .

$y + C_{1} = \int 49 e^{2 x} - 21 - 21 + 9 e^{- 2 x} d x$

Step 2.3.1.3.2

Subtract

$21$ from

$- 21$ .

$y + C_{1} = \int 49 e^{2 x} - 42 + 9 e^{- 2 x} d x$

Step 2.3.2

Split the single integral into multiple integrals.

$y + C_{1} = \int 49 e^{2 x} d x + \int - 42 d x + \int 9 e^{- 2 x} d x$

Step 2.3.3

Since

$49$ is constant with respect to

$x$ , move

$49$ out of the integral.

$y + C_{1} = 49 \int e^{2 x} d x + \int - 42 d x + \int 9 e^{- 2 x} d x$

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

$d u_{1}$ .

$y + C_{1} = 49 \int e^{u_{1}} \frac{1}{2} d u_{1} + \int - 42 d x + \int 9 e^{- 2 x} d x$

Step 2.3.5

Combine

$e^{u_{1}}$ and

$\frac{1}{2}$ .

$y + C_{1} = 49 \int \frac{e^{u_{1}}}{2} d u_{1} + \int - 42 d x + \int 9 e^{- 2 x} d x$

Step 2.3.6

Since

$\frac{1}{2}$ is constant with respect to

$u_{1}$ , move

$\frac{1}{2}$ out of the integral.

$y + C_{1} = 49 (\frac{1}{2} \int e^{u_{1}} d u_{1}) + \int - 42 d x + \int 9 e^{- 2 x} d x$

Step 2.3.7

Combine

$\frac{1}{2}$ and

$49$ .

$y + C_{1} = \frac{49}{2} \int e^{u_{1}} d u_{1} + \int - 42 d x + \int 9 e^{- 2 x} d x$

Step 2.3.8

The integral of

$e^{u_{1}}$ with respect to

$u_{1}$ is

$e^{u_{1}}$ .

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) + \int - 42 d x + \int 9 e^{- 2 x} d x$

Step 2.3.9

Apply the constant rule.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} + \int 9 e^{- 2 x} d x$

Step 2.3.10

Since

$9$ is constant with respect to

$x$ , move

$9$ out of the integral.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} + 9 \int e^{- 2 x} d x$

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

$d u_{2}$ .

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} + 9 \int e^{u_{2}} \frac{1}{- 2} d u_{2}$

Step 2.3.12

Simplify.

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Step 2.3.12.1

Move the negative in front of the fraction.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} + 9 \int e^{u_{2}} (- \frac{1}{2}) d u_{2}$

Step 2.3.12.2

Combine

$e^{u_{2}}$ and

$\frac{1}{2}$ .

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} + 9 \int - \frac{e^{u_{2}}}{2} d u_{2}$

Step 2.3.13

Since

$- 1$ is constant with respect to

$u_{2}$ , move

$- 1$ out of the integral.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} + 9 (- \int \frac{e^{u_{2}}}{2} d u_{2})$

Step 2.3.14

Multiply

$- 1$ by

$9$ .

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} - 9 \int \frac{e^{u_{2}}}{2} d u_{2}$

Step 2.3.15

Since

$\frac{1}{2}$ is constant with respect to

$u_{2}$ , move

$\frac{1}{2}$ out of the integral.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} - 9 (\frac{1}{2} \int e^{u_{2}} d u_{2})$

Step 2.3.16

Simplify.

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Step 2.3.16.1

Combine

$\frac{1}{2}$ and

$- 9$ .

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} + \frac{- 9}{2} \int e^{u_{2}} d u_{2}$

Step 2.3.16.2

Move the negative in front of the fraction.

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} - \frac{9}{2} \int e^{u_{2}} d u_{2}$

Step 2.3.17

The integral of

$e^{u_{2}}$ with respect to

$u_{2}$ is

$e^{u_{2}}$ .

$y + C_{1} = \frac{49}{2} (e^{u_{1}} + C_{2}) - 42 x + C_{3} - \frac{9}{2} (e^{u_{2}} + C_{4})$

Step 2.3.18

Simplify.

$y + C_{1} = \frac{49}{2} e^{u_{1}} - 42 x - \frac{9}{2} e^{u_{2}} + C_{5}$

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

$2 x$ .

$y + C_{1} = \frac{49}{2} e^{2 x} - 42 x - \frac{9}{2} e^{u_{2}} + C_{5}$

Step 2.3.19.2

Replace all occurrences of

$u_{2}$ with

$- 2 x$ .

$y + C_{1} = \frac{49}{2} e^{2 x} - 42 x - \frac{9}{2} e^{- 2 x} + C_{5}$

Step 2.3.20

Reorder terms.

$y + C_{1} = \frac{49}{2} e^{2 x} - \frac{9}{2} e^{- 2 x} - 42 x + C_{5}$

Step 2.4

Group the constant of integration on the right side as

$K$ .

$y = \frac{49}{2} e^{2 x} - \frac{9}{2} e^{- 2 x} - 42 x + K$