Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y + e^{y}$ .

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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Step 2.3.4.1

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

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Step 2.3.4.1.1

Differentiate $- x$ .

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

Step 2.3.4.1.2

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 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 $n = 1$ .

$- 1 \cdot 1$

Step 2.3.4.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.3.4.2

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.6.2

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

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

Step 2.3.7

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

$\frac{1}{2} y^{2} + e^{y} + C_{3} = \frac{1}{2} x^{2} + C_{4} + e^{u} + C_{5}$

Step 2.3.8

Simplify.

$\frac{1}{2} y^{2} + e^{y} + C_{3} = \frac{1}{2} x^{2} + e^{u} + C_{6}$

Step 2.3.9

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

$\frac{1}{2} y^{2} + e^{y} + C_{3} = \frac{1}{2} x^{2} + e^{- x} + C_{6}$

Step 2.4

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

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