Calculus Examples

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

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = x^{2} + y^{2} + x$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [x^{2} + y^{2} + x]$

Step 1.2

By the Sum Rule, the derivative of $x^{2} + y^{2} + x$ with respect to $y$ is $\frac{d}{d y} [x^{2}] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [x]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [x^{2}] + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [x]$

Step 1.3

Since $x^{2}$ is constant with respect to $y$ , the derivative of $x^{2}$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [y^{2}] + \frac{d}{d y} [x]$

Step 1.4

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

$\frac{\partial M}{\partial y} = 0 + 2 y + \frac{d}{d y} [x]$

Step 1.5

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

$\frac{\partial M}{\partial y} = 0 + 2 y + 0$

Step 1.6

Combine terms.

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Step 1.6.1

Add $0$ and $2 y$ .

$\frac{\partial M}{\partial y} = 2 y + 0$

Step 1.6.2

Add $2 y$ and $0$ .

$\frac{\partial M}{\partial y} = 2 y$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = y$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [y]$

Step 2.2

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

$\frac{\partial N}{\partial x} = 0$

Step 3

Check that $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

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Step 3.1

Substitute $2 y$ for $\frac{\partial M}{\partial y}$ and $0$ for $\frac{\partial N}{\partial x}$ .

$2 y = 0$

Step 3.2

Since the left side does not equal the right side, the equation is not an identity.

$2 y = 0$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

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Step 4.1

Substitute $2 y$ for $\frac{\partial M}{\partial y}$ .

$\frac{2 y - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $0$ for $\frac{\partial N}{\partial x}$ .

$\frac{2 y + 0}{N}$

Step 4.3

Substitute $y$ for $N$ .

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Step 4.3.1

Substitute $y$ for $N$ .

$\frac{2 y + 0}{y}$

Step 4.3.2

Add $2 y$ and $0$ .

$\frac{2 y}{y}$

Step 4.3.3

Cancel the common factor of $y$ .

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Step 4.3.3.1

Cancel the common factor.

$\frac{2 y}{y}$

Step 4.3.3.2

Divide $2$ by $1$ .

$2$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

$μ (x, y) = e^{\int 2 d x}$

Step 5

Evaluate the integral $e^{\int 2 d x}$ .

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Step 5.1

Apply the constant rule.

$μ (x, y) = e^{2 x + C}$

Step 5.2

Simplify.

$μ (x, y) = e^{2 x} + C$

Step 6

Multiply both sides of $(x^{2} + y^{2} + x) d x + y d y = 0$ by the integration factor $e^{2 x}$ .

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Step 6.1

Multiply $x^{2} + y^{2} + x$ by $e^{2 x}$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $y$ by $e^{2 x}$ .

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

Step 7

Set $f (x, y)$ equal to the integral of $N (x, y)$ .

$f (x, y) = \int y e^{2 x} d y$

Step 8

Integrate $N (x, y) = y e^{2 x}$ to find $f (x, y)$ .

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Step 8.1

Since $e^{2 x}$ is constant with respect to $y$ , move $e^{2 x}$ out of the integral.

$f (x, y) = e^{2 x} \int y d y$

Step 8.2

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

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

Step 8.3

Simplify the answer.

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Step 8.3.1

Rewrite $e^{2 x} (\frac{1}{2} y^{2} + C)$ as $e^{2 x} \frac{1}{2} y^{2} + C$ .

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

Step 8.3.2

Simplify.

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Step 8.3.2.1

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

$f (x, y) = \frac{e^{2 x}}{2} y^{2} + C$

Step 8.3.2.2

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

$f (x, y) = \frac{e^{2 x} y^{2}}{2} + C$

Step 8.3.3

Reorder terms.

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

Step 9

Since the integral of $g (x)$ will contain an integration constant, we can replace $C$ with $g (x)$ .

$f (x, y) = \frac{1}{2} e^{2 x} y^{2} + g (x)$

Step 10

Set $\frac{\partial f}{\partial x} = M (x, y)$ .

$\frac{\partial f}{\partial x} = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11

Find $\frac{\partial f}{\partial x}$ .

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

By the Sum Rule, the derivative of $\frac{1}{2} e^{2 x} y^{2} + g (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{2} e^{2 x} y^{2}] + \frac{d}{d x} [g (x)]$ .

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

Step 11.3

Evaluate $\frac{d}{d x} [\frac{1}{2} e^{2 x} y^{2}]$ .

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Step 11.3.1

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

$\frac{d}{d x} [\frac{e^{2 x}}{2} y^{2}] + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.2

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

$\frac{d}{d x} [\frac{e^{2 x} y^{2}}{2}] + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.3

Since $\frac{y^{2}}{2}$ is constant with respect to $x$ , the derivative of $\frac{e^{2 x} y^{2}}{2}$ with respect to $x$ is $\frac{y^{2}}{2} \frac{d}{d x} [e^{2 x}]$ .

$\frac{y^{2}}{2} \frac{d}{d x} [e^{2 x}] + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 11.3.4.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\frac{y^{2}}{2} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x]) + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.4.2

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

$\frac{y^{2}}{2} (e^{u} \frac{d}{d x} [2 x]) + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.4.3

Replace all occurrences of $u$ with $2 x$ .

$\frac{y^{2}}{2} (e^{2 x} \frac{d}{d x} [2 x]) + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{y^{2}}{2} (e^{2 x} (2 \frac{d}{d x} [x])) + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.6

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

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

Step 11.3.7

Multiply $2$ by $1$ .

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

Step 11.3.8

Move $2$ to the left of $e^{2 x}$ .

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

Step 11.3.9

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

$\frac{2 y^{2}}{2} e^{2 x} + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.10

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

$\frac{2 y^{2} e^{2 x}}{2} + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.11

Cancel the common factor of $2$ .

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Step 11.3.11.1

Cancel the common factor.

$\frac{2 y^{2} e^{2 x}}{2} + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.3.11.2

Divide $y^{2} e^{2 x}$ by $1$ .

$y^{2} e^{2 x} + \frac{d}{d x} [g (x)] = x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x}$

Step 11.4

Differentiate using the function rule which states that the derivative of $g (x)$ is $\frac{d g}{d x}$ .

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

Step 11.5

Simplify.

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Step 11.5.1

Reorder terms.

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

Step 11.5.2

Reorder factors in $\frac{d g}{d x} + e^{2 x} y^{2}$ .

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

Step 12

Solve for $\frac{d g}{d x}$ .

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Step 12.1

Move all terms not containing $\frac{d g}{d x}$ to the right side of the equation.

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Step 12.1.1

Subtract $y^{2} e^{2 x}$ from both sides of the equation.

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

Step 12.1.2

Combine the opposite terms in $x^{2} e^{2 x} + y^{2} e^{2 x} + x e^{2 x} - y^{2} e^{2 x}$ .

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Step 12.1.2.1

Subtract $y^{2} e^{2 x}$ from $y^{2} e^{2 x}$ .

$\frac{d g}{d x} = x^{2} e^{2 x} + x e^{2 x} + 0$

Step 12.1.2.2

Add $x^{2} e^{2 x} + x e^{2 x}$ and $0$ .

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

Step 13

Find the antiderivative of $x^{2} e^{2 x} + x e^{2 x}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = x^{2} e^{2 x} + x e^{2 x}$ .

$\int \frac{d g}{d x} d x = \int x^{2} e^{2 x} + x e^{2 x} d x$

Step 13.2

Evaluate $\int \frac{d g}{d x} d x$ .

$g (x) = \int x^{2} e^{2 x} + x e^{2 x} d x$

Step 13.3

Split the single integral into multiple integrals.

$g (x) = \int x^{2} e^{2 x} d x + \int x e^{2 x} d x$

Step 13.4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = e^{2 x}$ .

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

Step 13.5

Simplify.

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Step 13.5.1

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

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

Step 13.5.2

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

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

Step 13.6

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

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

Step 13.7

Simplify.

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Step 13.7.1

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{2}{2} \int e^{2 x} (x) d x) + \int x e^{2 x} d x$

Step 13.7.2

Cancel the common factor of $2$ .

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Step 13.7.2.1

Cancel the common factor.

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{2}{2} \int e^{2 x} (x) d x) + \int x e^{2 x} d x$

Step 13.7.2.2

Rewrite the expression.

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

Step 13.7.3

Multiply $\int e^{2 x} (x) d x$ by $1$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} - \int e^{2 x} (x) d x + \int x e^{2 x} d x$

Step 13.8

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{2 x}$ .

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

Step 13.9

Simplify.

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Step 13.9.1

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

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

Step 13.9.2

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

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

Step 13.9.3

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \int \frac{e^{2 x}}{2} d x) + \int x e^{2 x} d x$

Step 13.10

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

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

Step 13.11

Remove parentheses.

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

Step 13.12

Let $u_{1} = 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 13.12.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 13.12.1.1

Differentiate $2 x$ .

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

Step 13.12.1.2

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 13.12.1.3

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 13.12.1.4

Multiply $2$ by $1$ .

$2$

Step 13.12.2

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

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

Step 13.13

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

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

Step 13.14

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

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

Step 13.15

Simplify.

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Step 13.15.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{2 \cdot 2} \int e^{u_{1}} d u_{1}) + \int x e^{2 x} d x$

Step 13.15.2

Multiply $2$ by $2$ .

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

Step 13.16

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \int x e^{2 x} d x$

Step 13.17

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{2 x}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + x (\frac{1}{2} e^{2 x}) - \int \frac{1}{2} e^{2 x} d x$

Step 13.18

Simplify.

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Step 13.18.1

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + x \frac{e^{2 x}}{2} - \int \frac{1}{2} e^{2 x} d x$

Step 13.18.2

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \frac{x e^{2 x}}{2} - \int \frac{1}{2} e^{2 x} d x$

Step 13.19

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \frac{x e^{2 x}}{2} - (\frac{1}{2} \int e^{2 x} d x)$

Step 13.20

Remove parentheses.

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \frac{x e^{2 x}}{2} - \frac{1}{2} \int e^{2 x} d x$

Step 13.21

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 13.21.1

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

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Step 13.21.1.1

Differentiate $2 x$ .

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

Step 13.21.1.2

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 13.21.1.3

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 13.21.1.4

Multiply $2$ by $1$ .

$2$

Step 13.21.2

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \frac{x e^{2 x}}{2} - \frac{1}{2} \int e^{u_{2}} \frac{1}{2} d u_{2}$

Step 13.22

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \frac{x e^{2 x}}{2} - \frac{1}{2} \int \frac{e^{u_{2}}}{2} d u_{2}$

Step 13.23

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \frac{x e^{2 x}}{2} - \frac{1}{2} (\frac{1}{2} \int e^{u_{2}} d u_{2})$

Step 13.24

Simplify.

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Step 13.24.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \frac{x e^{2 x}}{2} - \frac{1}{2 \cdot 2} \int e^{u_{2}} d u_{2}$

Step 13.24.2

Multiply $2$ by $2$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \frac{x e^{2 x}}{2} - \frac{1}{4} \int e^{u_{2}} d u_{2}$

Step 13.25

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

$g (x) = \frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C)) + \frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{2}} + C)$

Step 13.26

Simplify.

$g (x) = \frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u_{1}}}{4} + \frac{x e^{2 x}}{2} - \frac{1}{4} e^{u_{2}} + C$

Step 13.27

Simplify.

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Step 13.27.1

Add $- \frac{x e^{2 x}}{2}$ and $\frac{x e^{2 x}}{2}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + 0 + \frac{e^{u_{1}}}{4} - \frac{1}{4} e^{u_{2}} + C$

Step 13.27.2

Add $\frac{x^{2} e^{2 x}}{2}$ and $0$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}}}{4} - \frac{1}{4} e^{u_{2}} + C$

Step 13.27.3

To write $- \frac{1}{4} e^{u_{2}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}}}{4} - \frac{1}{4} e^{u_{2}} \cdot \frac{4}{4} + C$

Step 13.27.4

Combine $- \frac{1}{4} e^{u_{2}}$ and $\frac{4}{4}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}}}{4} + \frac{- \frac{1}{4} e^{u_{2}} \cdot 4}{4} + C$

Step 13.27.5

Combine the numerators over the common denominator.

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}} - \frac{1}{4} e^{u_{2}} \cdot 4}{4} + C$

Step 13.27.6

Combine $e^{u_{2}}$ and $\frac{1}{4}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}} - \frac{e^{u_{2}}}{4} \cdot 4}{4} + C$

Step 13.27.7

Multiply $4$ by $- 1$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}} - 4 \frac{e^{u_{2}}}{4}}{4} + C$

Step 13.27.8

Combine $- 4$ and $\frac{e^{u_{2}}}{4}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}} + \frac{- 4 e^{u_{2}}}{4}}{4} + C$

Step 13.27.9

Cancel the common factor of $- 4$ and $4$ .

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Step 13.27.9.1

Factor $4$ out of $- 4 e^{u_{2}}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}} + \frac{4 (- e^{u_{2}})}{4}}{4} + C$

Step 13.27.9.2

Cancel the common factors.

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Step 13.27.9.2.1

Factor $4$ out of $4$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}} + \frac{4 (- e^{u_{2}})}{4 (1)}}{4} + C$

Step 13.27.9.2.2

Cancel the common factor.

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}} + \frac{4 (- e^{u_{2}})}{4 \cdot 1}}{4} + C$

Step 13.27.9.2.3

Rewrite the expression.

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}} + \frac{- e^{u_{2}}}{1}}{4} + C$

Step 13.27.9.2.4

Divide $- e^{u_{2}}$ by $1$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{e^{u_{1}} - e^{u_{2}}}{4} + C$

Step 13.27.10

Subtract $e^{u_{2}}$ from $e^{u_{1}}$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{0}{4} + C$

Step 13.27.11

Cancel the common factor of $0$ and $4$ .

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Step 13.27.11.1

Factor $4$ out of $0$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{4 (0)}{4} + C$

Step 13.27.11.2

Cancel the common factors.

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Step 13.27.11.2.1

Factor $4$ out of $4$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{4 \cdot 0}{4 \cdot 1} + C$

Step 13.27.11.2.2

Cancel the common factor.

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{4 \cdot 0}{4 \cdot 1} + C$

Step 13.27.11.2.3

Rewrite the expression.

$g (x) = \frac{x^{2} e^{2 x}}{2} + \frac{0}{1} + C$

Step 13.27.11.2.4

Divide $0$ by $1$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + 0 + C$

Step 13.27.12

Add $\frac{x^{2} e^{2 x}}{2}$ and $0$ .

$g (x) = \frac{x^{2} e^{2 x}}{2} + C$

Step 13.28

Reorder terms.

$g (x) = \frac{1}{2} x^{2} e^{2 x} + C$

Step 14

Substitute for $g (x)$ in $f (x, y) = \frac{1}{2} e^{2 x} y^{2} + g (x)$ .

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

Step 15

Simplify $f (x, y) = \frac{1}{2} e^{2 x} y^{2} + \frac{1}{2} x^{2} e^{2 x} + C$ .

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Step 15.1

Simplify each term.

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Step 15.1.1

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

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

Step 15.1.2

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

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

Step 15.1.3

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

$f (x, y) = \frac{e^{2 x} y^{2}}{2} + \frac{x^{2}}{2} e^{2 x} + C$

Step 15.1.4

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

$f (x, y) = \frac{e^{2 x} y^{2}}{2} + \frac{x^{2} e^{2 x}}{2} + C$

Step 15.2

Reorder factors in $f (x, y) = \frac{e^{2 x} y^{2}}{2} + \frac{x^{2} e^{2 x}}{2} + C$ .

$f (x, y) = \frac{y^{2} e^{2 x}}{2} + \frac{x^{2} e^{2 x}}{2} + C$