Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y + y e^{y^{2}}$ .

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

Step 1.2

Simplify.

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Step 1.2.1

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

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Step 1.2.1.1

Raise $y$ to the power of $1$ .

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

Step 1.2.1.2

Factor $y$ out of $y^{1}$ .

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Factor $y$ out of $y \cdot 1 + y (e^{y^{2}})$ .

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

Step 1.2.2

Multiply $y + y e^{y^{2}}$ by $\frac{1 + x e^{x}}{y (1 + e^{y^{2}})}$ .

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

Step 1.2.3

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

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Step 1.2.3.1

Raise $y$ to the power of $1$ .

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

Step 1.2.3.2

Factor $y$ out of $y^{1}$ .

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

Step 1.2.3.3

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

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

Step 1.2.3.4

Factor $y$ out of $y \cdot 1 + y (e^{y^{2}})$ .

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

Step 1.2.4

Cancel the common factor of $y$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.2.5

Cancel the common factor of $1 + e^{y^{2}}$ .

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Step 1.2.5.1

Cancel the common factor.

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

Step 1.2.5.2

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

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

Let $u = y^{2}$ . Then $d u = 2 y d y$ , so $\frac{1}{2} d u = y d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.3.1

Let $u = y^{2}$ . Find $\frac{d u}{d y}$ .

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Step 2.2.3.1.1

Differentiate $y^{2}$ .

$\frac{d}{d y} [y^{2}]$

Step 2.2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$2 y$

Step 2.2.3.2

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

Simplify.

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

Step 2.2.8

Replace all occurrences of $u$ with $y^{2}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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

Step 2.3.6

Reorder terms.

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

Step 2.4

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

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