Calculus Examples

${(e^{- y} + 1)}^{- 2} e^{x} d x + {(e^{- x} + 1)}^{- 3} e^{y} d y = 0$

Step 1

Subtract ${(e^{- y} + 1)}^{- 2} e^{x} d x$ from both sides of the equation.

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

Step 2

Multiply both sides by ${(e^{- x} + 1)}^{3} {(e^{- y} + 1)}^{2}$ .

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

Step 3

Simplify.

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Step 3.1

Multiply ${(e^{- x} + 1)}^{3}$ by ${(e^{- x} + 1)}^{- 3}$ by adding the exponents.

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Step 3.1.1

Move ${(e^{- x} + 1)}^{- 3}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Add $- 3$ and $3$ .

${(e^{- x} + 1)}^{0} {(e^{- y} + 1)}^{2} e^{y} d y = {(e^{- x} + 1)}^{3} {(e^{- y} + 1)}^{2} (- {(e^{- y} + 1)}^{- 2} e^{x}) d x$

Step 3.2

Simplify ${(e^{- x} + 1)}^{0} {(e^{- y} + 1)}^{2} e^{y}$ .

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

Step 3.3

Rewrite ${(e^{- y} + 1)}^{2}$ as $(e^{- y} + 1) (e^{- y} + 1)$ .

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

Step 3.4

Expand $(e^{- y} + 1) (e^{- y} + 1)$ using the FOIL Method.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Apply the distributive property.

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

Step 3.4.3

Apply the distributive property.

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

Step 3.5

Simplify and combine like terms.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

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

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Step 3.5.1.1.1

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

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

Step 3.5.1.1.2

Subtract $y$ from $- y$ .

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

Step 3.5.1.2

Multiply $e^{- y}$ by $1$ .

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

Step 3.5.1.3

Multiply $e^{- y}$ by $1$ .

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

Step 3.5.1.4

Multiply $1$ by $1$ .

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

Step 3.5.2

Add $e^{- y}$ and $e^{- y}$ .

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Simplify.

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Step 3.7.1

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

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Step 3.7.1.1

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

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

Step 3.7.1.2

Add $- 2 y$ and $y$ .

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

Step 3.7.2

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

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Step 3.7.2.1

Move $e^{y}$ .

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

Step 3.7.2.2

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

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

Step 3.7.2.3

Subtract $y$ from $y$ .

$(e^{- y} + 2 e^{0} + 1 e^{y}) d y = {(e^{- x} + 1)}^{3} {(e^{- y} + 1)}^{2} (- {(e^{- y} + 1)}^{- 2} e^{x}) d x$

Step 3.7.3

Simplify $2 e^{0}$ .

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

Step 3.7.4

Multiply $e^{y}$ by $1$ .

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

Step 3.8

Rewrite using the commutative property of multiplication.

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

Step 3.9

Multiply ${(e^{- y} + 1)}^{2}$ by ${(e^{- y} + 1)}^{- 2}$ by adding the exponents.

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Step 3.9.1

Move ${(e^{- y} + 1)}^{- 2}$ .

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

Step 3.9.2

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

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

Step 3.9.3

Add $- 2$ and $2$ .

$(e^{- y} + 2 + e^{y}) d y = - {(e^{- x} + 1)}^{3} {(e^{- y} + 1)}^{0} e^{x} d x$

Step 3.10

Simplify $- {(e^{- x} + 1)}^{3} {(e^{- y} + 1)}^{0} e^{x}$ .

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

Step 3.11

Use the Binomial Theorem.

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

Step 3.12

Simplify each term.

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Step 3.12.1

Multiply the exponents in ${(e^{- x})}^{3}$ .

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Step 3.12.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.12.1.2

Multiply $3$ by $- 1$ .

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

Step 3.12.2

Multiply the exponents in ${(e^{- x})}^{2}$ .

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Step 3.12.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.12.2.2

Multiply $2$ by $- 1$ .

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

Step 3.12.3

Multiply $3$ by $1$ .

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

Step 3.12.4

One to any power is one.

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

Step 3.12.5

Multiply $3$ by $1$ .

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

Step 3.12.6

One to any power is one.

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

Step 3.13

Apply the distributive property.

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

Step 3.14

Simplify.

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Step 3.14.1

Multiply $3$ by $- 1$ .

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

Step 3.14.2

Multiply $3$ by $- 1$ .

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

Step 3.14.3

Multiply $- 1$ by $1$ .

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

Step 3.15

Apply the distributive property.

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

Step 3.16

Simplify.

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Step 3.16.1

Multiply $e^{- 3 x}$ by $e^{x}$ by adding the exponents.

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Step 3.16.1.1

Move $e^{x}$ .

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

Step 3.16.1.2

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

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

Step 3.16.1.3

Subtract $3 x$ from $x$ .

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

Step 3.16.2

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

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Step 3.16.2.1

Move $e^{x}$ .

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

Step 3.16.2.2

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

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

Step 3.16.2.3

Subtract $2 x$ from $x$ .

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

Step 3.16.3

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

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Step 3.16.3.1

Move $e^{x}$ .

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

Step 3.16.3.2

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

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

Step 3.16.3.3

Subtract $x$ from $x$ .

$(e^{- y} + 2 + e^{y}) d y = (- e^{- 2 x} - 3 e^{- x} - 3 e^{0} - 1 e^{x}) d x$

Step 3.16.4

Simplify $- 3 e^{0}$ .

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

Step 3.16.5

Rewrite $- 1 e^{x}$ as $- e^{x}$ .

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int e^{- y} + 2 + e^{y} d y = \int - e^{- 2 x} - 3 e^{- x} - 3 - e^{x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Split the single integral into multiple integrals.

$\int e^{- y} d y + \int 2 d y + \int e^{y} d y = \int - e^{- 2 x} - 3 e^{- x} - 3 - e^{x} d x$

Step 4.2.2

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

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Step 4.2.2.1

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

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Step 4.2.2.1.1

Differentiate $- y$ .

$\frac{d}{d y} [- y]$

Step 4.2.2.1.2

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

$- \frac{d}{d y} [y]$

Step 4.2.2.1.3

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

$- 1 \cdot 1$

Step 4.2.2.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.2.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Apply the constant rule.

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

Step 4.2.6

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

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

Step 4.2.7

Simplify.

$- e^{u_{1}} + 2 y + e^{y} + C_{4} = \int - e^{- 2 x} - 3 e^{- x} - 3 - e^{x} d x$

Step 4.2.8

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

$- e^{- y} + 2 y + e^{y} + C_{4} = \int - e^{- 2 x} - 3 e^{- x} - 3 - e^{x} d x$

Step 4.2.9

Reorder terms.

$2 y + e^{y} - e^{- y} + C_{4} = \int - e^{- 2 x} - 3 e^{- x} - 3 - e^{x} d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

Split the single integral into multiple integrals.

$2 y + e^{y} - e^{- y} + C_{4} = \int - e^{- 2 x} d x + \int - 3 e^{- x} d x + \int - 3 d x + \int - e^{x} d x$

Step 4.3.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$2 y + e^{y} - e^{- y} + C_{4} = - \int e^{- 2 x} d x + \int - 3 e^{- x} d x + \int - 3 d x + \int - e^{x} d x$

Step 4.3.3

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 4.3.3.1

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

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Step 4.3.3.1.1

Differentiate $- 2 x$ .

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

Step 4.3.3.1.2

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 4.3.3.1.3

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 4.3.3.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 4.3.3.2

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

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

Step 4.3.4

Simplify.

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Step 4.3.4.1

Move the negative in front of the fraction.

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

Step 4.3.4.2

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

$2 y + e^{y} - e^{- y} + C_{4} = - \int - \frac{e^{u_{2}}}{2} d u_{2} + \int - 3 e^{- x} d x + \int - 3 d x + \int - e^{x} d x$

Step 4.3.5

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

$2 y + e^{y} - e^{- y} + C_{4} = - - \int \frac{e^{u_{2}}}{2} d u_{2} + \int - 3 e^{- x} d x + \int - 3 d x + \int - e^{x} d x$

Step 4.3.6

Simplify.

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Step 4.3.6.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.6.2

Multiply $\int \frac{e^{u_{2}}}{2} d u_{2}$ by $1$ .

$2 y + e^{y} - e^{- y} + C_{4} = \int \frac{e^{u_{2}}}{2} d u_{2} + \int - 3 e^{- x} d x + \int - 3 d x + \int - e^{x} d x$

Step 4.3.7

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

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

Step 4.3.8

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

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} (e^{u_{2}} + C_{5}) + \int - 3 e^{- x} d x + \int - 3 d x + \int - e^{x} d x$

Step 4.3.9

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} (e^{u_{2}} + C_{5}) - 3 \int e^{- x} d x + \int - 3 d x + \int - e^{x} d x$

Step 4.3.10

Let $u_{3} = - x$ . Then $d u_{3} = - d x$ , so $- d u_{3} = d x$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 4.3.10.1

Let $u_{3} = - x$ . Find $\frac{d u_{3}}{d x}$ .

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Step 4.3.10.1.1

Differentiate $- x$ .

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

Step 4.3.10.1.2

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

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

Step 4.3.10.1.3

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

$- 1 \cdot 1$

Step 4.3.10.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.3.10.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} (e^{u_{2}} + C_{5}) - 3 \int - e^{u_{3}} d u_{3} + \int - 3 d x + \int - e^{x} d x$

Step 4.3.11

Since $- 1$ is constant with respect to $u_{3}$ , move $- 1$ out of the integral.

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} (e^{u_{2}} + C_{5}) - 3 (- \int e^{u_{3}} d u_{3}) + \int - 3 d x + \int - e^{x} d x$

Step 4.3.12

Multiply $- 1$ by $- 3$ .

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} (e^{u_{2}} + C_{5}) + 3 \int e^{u_{3}} d u_{3} + \int - 3 d x + \int - e^{x} d x$

Step 4.3.13

The integral of $e^{u_{3}}$ with respect to $u_{3}$ is $e^{u_{3}}$ .

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} (e^{u_{2}} + C_{5}) + 3 (e^{u_{3}} + C_{6}) + \int - 3 d x + \int - e^{x} d x$

Step 4.3.14

Apply the constant rule.

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} (e^{u_{2}} + C_{5}) + 3 (e^{u_{3}} + C_{6}) - 3 x + C_{7} + \int - e^{x} d x$

Step 4.3.15

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} (e^{u_{2}} + C_{5}) + 3 (e^{u_{3}} + C_{6}) - 3 x + C_{7} - \int e^{x} d x$

Step 4.3.16

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

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} (e^{u_{2}} + C_{5}) + 3 (e^{u_{3}} + C_{6}) - 3 x + C_{7} - (e^{x} + C_{8})$

Step 4.3.17

Simplify.

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

Step 4.3.18

Substitute back in for each integration substitution variable.

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Step 4.3.18.1

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

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

Step 4.3.18.2

Replace all occurrences of $u_{3}$ with $- x$ .

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} e^{- 2 x} + 3 e^{- x} - 3 x - e^{x} + C_{9}$

Step 4.3.19

Reorder terms.

$2 y + e^{y} - e^{- y} + C_{4} = \frac{1}{2} e^{- 2 x} - 3 x + 3 e^{- x} - e^{x} + C_{9}$

Step 4.4

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

$2 y + e^{y} - e^{- y} = \frac{1}{2} e^{- 2 x} - 3 x + 3 e^{- x} - e^{x} + K$