Calculus Examples

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

Step 1

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

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

Step 2

Multiply both sides by $\frac{1}{e^{x}}$ .

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

Cancel the common factor of $e^{x}$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{e^{x}}$ into the numerator.

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

Step 3.2.2

Factor $e^{x}$ out of $e^{x} y^{2}$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Multiply $- 1$ by $1$ .

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

Step 3.6

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

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

Step 3.7

Cancel the common factor of $e^{2 x}$ and $e^{x}$ .

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Step 3.7.1

Factor $e^{x}$ out of $e^{2 x}$ .

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

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Multiply by $1$ .

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

Step 3.7.2.2

Cancel the common factor.

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

Step 3.7.2.3

Rewrite the expression.

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

Step 3.7.2.4

Divide $e^{x}$ by $1$ .

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

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

Step 4.2.3

Rewrite $- (\frac{1}{3} y^{3} + C_{1})$ as $- \frac{1}{3} y^{3} + C_{1}$ .

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Split the single integral into multiple integrals.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify the expression.

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Step 4.3.3.1

Negate the exponent of $e^{x}$ and move it out of the denominator.

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

Step 4.3.3.2

Simplify.

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Step 4.3.3.2.1

Multiply the exponents in ${(e^{x})}^{- 1}$ .

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Step 4.3.3.2.1.1

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

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

Step 4.3.3.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 4.3.3.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 4.3.3.2.2

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

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

Step 4.3.4

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.3.4.1

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

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Step 4.3.4.1.1

Differentiate $- x$ .

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

Step 4.3.4.1.2

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 4.3.4.1.3

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

$- 1 \cdot 1$

Step 4.3.4.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.3.4.2

Rewrite the problem using $u$ and $d u$ .

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

Step 4.3.5

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

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

Step 4.3.6

Simplify.

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Step 4.3.6.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.6.2

Multiply $\int e^{u} d u$ by $1$ .

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

Step 4.3.7

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

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

Step 4.3.8

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

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

Step 4.3.9

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

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

Step 4.3.10

Simplify.

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

Step 4.3.11

Replace all occurrences of $u$ with $- x$ .

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $- 3$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $- 3 (- \frac{1}{3} y^{3})$ .

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Step 5.2.1.1.1

Combine $y^{3}$ and $\frac{1}{3}$ .

$- 3 (- \frac{y^{3}}{3}) = - 3 (e^{- x} - e^{x} + K)$

Step 5.2.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.2.1

Move the leading negative in $- \frac{y^{3}}{3}$ into the numerator.

$- 3 \frac{- y^{3}}{3} = - 3 (e^{- x} - e^{x} + K)$

Step 5.2.1.1.2.2

Factor $3$ out of $- 3$ .

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

Step 5.2.1.1.2.3

Cancel the common factor.

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

Step 5.2.1.1.2.4

Rewrite the expression.

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

Step 5.2.1.1.3

Multiply.

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Step 5.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.1.1.3.2

Multiply $y^{3}$ by $1$ .

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $- 3 (e^{- x} - e^{x} + K)$ .

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Step 5.2.2.1.1

Apply the distributive property.

$y^{3} = - 3 e^{- x} - 3 (- e^{x}) - 3 K$

Step 5.2.2.1.2

Multiply $- 1$ by $- 3$ .

$y^{3} = - 3 e^{- x} + 3 e^{x} - 3 K$

Step 5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{- 3 e^{- x} + 3 e^{x} - 3 K}$

Step 5.4

Factor $3$ out of $- 3 e^{- x} + 3 e^{x} - 3 K$ .

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Step 5.4.1

Factor $3$ out of $- 3 e^{- x}$ .

$y = \sqrt[3]{3 (- e^{- x}) + 3 e^{x} - 3 K}$

Step 5.4.2

Factor $3$ out of $3 e^{x}$ .

$y = \sqrt[3]{3 (- e^{- x}) + 3 (e^{x}) - 3 K}$

Step 5.4.3

Factor $3$ out of $- 3 K$ .

$y = \sqrt[3]{3 (- e^{- x}) + 3 (e^{x}) + 3 (- K)}$

Step 5.4.4

Factor $3$ out of $3 (- e^{- x}) + 3 (e^{x})$ .

$y = \sqrt[3]{3 (- e^{- x} + e^{x}) + 3 (- K)}$

Step 5.4.5

Factor $3$ out of $3 (- e^{- x} + e^{x}) + 3 (- K)$ .

$y = \sqrt[3]{3 (- e^{- x} + e^{x} - K)}$

Step 6

Simplify the constant of integration.

$y = \sqrt[3]{3 (- e^{- x} + e^{x} + K)}$