Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Divide each term in $(1 + e^{x}) y \cdot \frac{d y}{d x} = e^{x}$ by $(1 + e^{x}) y$ and simplify.

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Step 1.1.1

Divide each term in $(1 + e^{x}) y \cdot \frac{d y}{d x} = e^{x}$ by $(1 + e^{x}) y$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $1 + e^{x}$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

Step 1.1.2.2

Cancel the common factor of $y$ .

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Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Reorder factors in $\frac{e^{x}}{(1 + e^{x}) y}$ .

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

Step 1.2

Regroup factors.

$\frac{d y}{d x} = \frac{e^{x}}{1 + e^{x}} \cdot \frac{1}{y}$

Step 1.3

Multiply both sides by $y$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Multiply $\frac{e^{x}}{1 + e^{x}}$ by $\frac{1}{y}$ .

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

Step 1.4.2

Cancel the common factor of $y$ .

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Step 1.4.2.1

Factor $y$ out of $(1 + e^{x}) y$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{e^{x}}{1 + e^{x}}$

Step 1.5

Rewrite the equation.

$y d y = \frac{e^{x}}{1 + e^{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 1 + e^{x}$ . Then $d u = e^{x} d x$ , so $\frac{1}{e^{x}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = 1 + e^{x}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $1 + e^{x}$ .

$\frac{d}{d x} [1 + e^{x}]$

Step 2.3.1.1.2

Differentiate.

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Step 2.3.1.1.2.1

By the Sum Rule, the derivative of $1 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [e^{x}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [e^{x}]$

Step 2.3.1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [e^{x}]$

Step 2.3.1.1.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$0 + e^{x}$

Step 2.3.1.1.4

Add $0$ and $e^{x}$ .

$e^{x}$

Step 2.3.1.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{u} d u$

Step 2.3.2

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\frac{1}{2} y^{2} + C_{1} = \ln (| u |) + C_{2}$

Step 2.3.3

Replace all occurrences of $u$ with $1 + e^{x}$ .

$\frac{1}{2} y^{2} + C_{1} = \ln (| 1 + e^{x} |) + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = \ln (| 1 + e^{x} |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\ln (| 1 + e^{x} |) + C)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (\ln (| 1 + e^{x} |) + C)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (\ln (| 1 + e^{x} |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\ln (| 1 + e^{x} |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$y^{2} = 2 \ln (| 1 + e^{x} |) + 2 C$

Step 3.3

Simplify $2 \ln (| 1 + e^{x} |)$ by moving $2$ inside the logarithm.

$y^{2} = \ln ({| 1 + e^{x} |}^{2}) + 2 C$

Step 3.4

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

$y = \pm \sqrt{\ln ({| 1 + e^{x} |}^{2}) + 2 C}$

Step 3.5

Remove the absolute value in ${| 1 + e^{x} |}^{2}$ because exponentiations with even powers are always positive.

$y = \pm \sqrt{\ln ({(1 + e^{x})}^{2}) + 2 C}$

Step 3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{\ln ({(1 + e^{x})}^{2}) + 2 C}$

Step 3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{\ln ({(1 + e^{x})}^{2}) + 2 C}$

Step 3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{\ln ({(1 + e^{x})}^{2}) + 2 C}$

$y = - \sqrt{\ln ({(1 + e^{x})}^{2}) + 2 C}$

$y = \sqrt{\ln ({(1 + e^{x})}^{2}) + 2 C}$

$y = - \sqrt{\ln ({(1 + e^{x})}^{2}) + 2 C}$

$y = \sqrt{\ln ({(1 + e^{x})}^{2}) + 2 C}$

$y = - \sqrt{\ln ({(1 + e^{x})}^{2}) + 2 C}$

Step 4

Simplify the constant of integration.

$y = \sqrt{\ln ({(1 + e^{x})}^{2}) + C}$

$y = - \sqrt{\ln ({(1 + e^{x})}^{2}) + C}$