Calculus Examples

$(x y^{2} + x^{2} y^{2} + 3) d x + (x^{2} y) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = x y^{2} + x^{2} y^{2} + 3$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [x y^{2} + x^{2} y^{2} + 3]$

Step 1.2

By the Sum Rule, the derivative of $x y^{2} + x^{2} y^{2} + 3$ with respect to $y$ is $\frac{d}{d y} [x y^{2}] + \frac{d}{d y} [x^{2} y^{2}] + \frac{d}{d y} [3]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [x y^{2}] + \frac{d}{d y} [x^{2} y^{2}] + \frac{d}{d y} [3]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = x \frac{d}{d y} [y^{2}] + \frac{d}{d y} [x^{2} y^{2}] + \frac{d}{d y} [3]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = x (2 y) + \frac{d}{d y} [x^{2} y^{2}] + \frac{d}{d y} [3]$

Step 1.3.3

Move $2$ to the left of $x$ .

$\frac{\partial M}{\partial y} = 2 x y + \frac{d}{d y} [x^{2} y^{2}] + \frac{d}{d y} [3]$

Step 1.4

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

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Step 1.4.1

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

$\frac{\partial M}{\partial y} = 2 x y + x^{2} \frac{d}{d y} [y^{2}] + \frac{d}{d y} [3]$

Step 1.4.2

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

$\frac{\partial M}{\partial y} = 2 x y + x^{2} (2 y) + \frac{d}{d y} [3]$

Step 1.4.3

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

$\frac{\partial M}{\partial y} = 2 x y + 2 x^{2} y + \frac{d}{d y} [3]$

Step 1.5

Differentiate using the Constant Rule.

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Step 1.5.1

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

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

Step 1.5.2

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

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

$\frac{\partial N}{\partial x} = y (2 x)$

Step 2.4

Reorder.

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Step 2.4.1

Move $2$ to the left of $y$ .

$\frac{\partial N}{\partial x} = 2 \cdot y x$

Step 2.4.2

Reorder the factors of $2 y x$ .

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

Step 3

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

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Step 3.1

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

$2 x y + 2 x^{2} y = 2 x y$

Step 3.2

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

$2 x y + 2 x^{2} y = 2 x y$ 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 $2 x y + 2 x^{2} y$ for $\frac{\partial M}{\partial y}$ .

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

Step 4.2

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

$\frac{2 x y + 2 x^{2} y - (2 x y)}{N}$

Step 4.3

Substitute $x^{2} y$ for $N$ .

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Step 4.3.1

Substitute $x^{2} y$ for $N$ .

$\frac{2 x y + 2 x^{2} y - (2 x y)}{x^{2} y}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $x y$ out of $2 x y + 2 x^{2} y - 1 \cdot 2 x y$ .

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Step 4.3.2.1.1

Factor $x y$ out of $2 x y$ .

$\frac{x y (2) + 2 x^{2} y - 1 \cdot 2 x y}{x^{2} y}$

Step 4.3.2.1.2

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

$\frac{x y (2) + x y (2 x) - 1 \cdot 2 x y}{x^{2} y}$

Step 4.3.2.1.3

Factor $x y$ out of $- 1 \cdot 2 x y$ .

$\frac{x y (2) + x y (2 x) + x y (- 1 \cdot 2)}{x^{2} y}$

Step 4.3.2.1.4

Factor $x y$ out of $x y (2) + x y (2 x)$ .

$\frac{x y (2 + 2 x) + x y (- 1 \cdot 2)}{x^{2} y}$

Step 4.3.2.1.5

Factor $x y$ out of $x y (2 + 2 x) + x y (- 1 \cdot 2)$ .

$\frac{x y (2 + 2 x - 1 \cdot 2)}{x^{2} y}$

Step 4.3.2.2

Multiply $- 1$ by $2$ .

$\frac{x y (2 + 2 x - 2)}{x^{2} y}$

Step 4.3.2.3

Combine the opposite terms in $2 + 2 x - 2$ .

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Step 4.3.2.3.1

Subtract $2$ from $2$ .

$\frac{x y (2 x + 0)}{x^{2} y}$

Step 4.3.2.3.2

Add $2 x$ and $0$ .

$\frac{x y (2 x)}{x^{2} y}$

$\frac{x y \cdot 2 x}{x^{2} y}$

Step 4.3.2.4

Combine exponents.

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Step 4.3.2.4.1

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

$\frac{x^{1} x y \cdot 2}{x^{2} y}$

Step 4.3.2.4.2

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

$\frac{x^{1} x^{1} y \cdot 2}{x^{2} y}$

Step 4.3.2.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} y \cdot 2}{x^{2} y}$

Step 4.3.2.4.4

Add $1$ and $1$ .

$\frac{x^{2} y \cdot 2}{x^{2} y}$

Step 4.3.3

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

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Step 4.3.3.1

Cancel the common factor.

$\frac{x^{2} y \cdot 2}{x^{2} y}$

Step 4.3.3.2

Rewrite the expression.

$\frac{y \cdot 2}{y}$

Step 4.3.4

Cancel the common factor of $y$ .

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Step 4.3.4.1

Cancel the common factor.

$\frac{y \cdot 2}{y}$

Step 4.3.4.2

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 $(x y^{2} + x^{2} y^{2} + 3) d x + x^{2} y d y = 0$ by the integration factor $e^{2 x}$ .

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Step 6.1

Multiply $x y^{2} + x^{2} y^{2} + 3$ by $e^{2 x}$ .

$(x y^{2} + x^{2} y^{2} + 3) e^{2 x} d x + x^{2} y d y = 0$

Step 6.2

Apply the distributive property.

$(x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}) d x + x^{2} y d y = 0$

Step 6.3

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

$(x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}) d x + x^{2} y e^{2 x} d y = 0$

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Simplify the answer.

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Step 8.3.1

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

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

Step 8.3.2

Simplify.

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Step 8.3.2.1

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

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

Step 8.3.2.2

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

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

Step 8.3.2.3

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

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

Step 8.3.3

Reorder terms.

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [\frac{1}{2} x^{2} e^{2 x} y^{2} + g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.2

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

$\frac{d}{d x} [\frac{1}{2} x^{2} e^{2 x} y^{2}] + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3

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

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Step 11.3.1

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

$\frac{d}{d x} [\frac{x^{2}}{2} e^{2 x} y^{2}] + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.2

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

$\frac{d}{d x} [\frac{x^{2} e^{2 x}}{2} y^{2}] + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.3

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

$\frac{d}{d x} [\frac{x^{2} e^{2 x} y^{2}}{2}] + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.4

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

$\frac{y^{2}}{2} \frac{d}{d x} [x^{2} e^{2 x}] + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = e^{2 x}$ .

$\frac{y^{2}}{2} (x^{2} \frac{d}{d x} [e^{2 x}] + e^{2 x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.6

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 11.3.6.1

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

$\frac{y^{2}}{2} (x^{2} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x]) + e^{2 x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.6.2

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

$\frac{y^{2}}{2} (x^{2} (e^{u} \frac{d}{d x} [2 x]) + e^{2 x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.6.3

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

$\frac{y^{2}}{2} (x^{2} (e^{2 x} \frac{d}{d x} [2 x]) + e^{2 x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.7

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

$\frac{y^{2}}{2} (x^{2} (e^{2 x} (2 \frac{d}{d x} [x])) + e^{2 x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.8

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

$\frac{y^{2}}{2} (x^{2} (e^{2 x} (2 \cdot 1)) + e^{2 x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.9

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

$\frac{y^{2}}{2} (x^{2} (e^{2 x} (2 \cdot 1)) + e^{2 x} (2 x)) + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.10

Multiply $2$ by $1$ .

$\frac{y^{2}}{2} (x^{2} (e^{2 x} \cdot 2) + e^{2 x} (2 x)) + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.3.11

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

$\frac{y^{2}}{2} (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + \frac{d}{d x} [g (x)] = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.4

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

$\frac{y^{2}}{2} (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5

Simplify.

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Step 11.5.1

Apply the distributive property.

$\frac{y^{2}}{2} (x^{2} (2 e^{2 x})) + \frac{y^{2}}{2} (e^{2 x} (2 x)) + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2

Combine terms.

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Step 11.5.2.1

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

$\frac{x^{2} y^{2}}{2} (2 e^{2 x}) + \frac{y^{2}}{2} (e^{2 x} (2 x)) + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2.2

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

$\frac{2 (x^{2} y^{2})}{2} e^{2 x} + \frac{y^{2}}{2} (e^{2 x} (2 x)) + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2.3

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

$\frac{2 (x^{2} y^{2}) e^{2 x}}{2} + \frac{y^{2}}{2} (e^{2 x} (2 x)) + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2.4

Cancel the common factor of $2$ .

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Step 11.5.2.4.1

Cancel the common factor.

$\frac{2 x^{2} y^{2} e^{2 x}}{2} + \frac{y^{2}}{2} (e^{2 x} (2 x)) + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2.4.2

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

$x^{2} y^{2} e^{2 x} + \frac{y^{2}}{2} (e^{2 x} (2 x)) + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2.5

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

$x^{2} y^{2} e^{2 x} + \frac{e^{2 x} y^{2}}{2} (2 x) + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2.6

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

$x^{2} y^{2} e^{2 x} + \frac{2 (e^{2 x} y^{2})}{2} x + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2.7

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

$x^{2} y^{2} e^{2 x} + \frac{2 (e^{2 x} y^{2}) x}{2} + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2.8

Cancel the common factor of $2$ .

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Step 11.5.2.8.1

Cancel the common factor.

$x^{2} y^{2} e^{2 x} + \frac{2 e^{2 x} y^{2} x}{2} + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.2.8.2

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

$x^{2} y^{2} e^{2 x} + e^{2 x} y^{2} x + \frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.3

Reorder terms.

$\frac{d g}{d x} + e^{2 x} x^{2} y^{2} + e^{2 x} y^{2} x = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

Step 11.5.4

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

$\frac{d g}{d x} + x^{2} y^{2} e^{2 x} + y^{2} x e^{2 x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x}$

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

$\frac{d g}{d x} + y^{2} x e^{2 x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x} - x^{2} y^{2} e^{2 x}$

Step 12.1.2

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

$\frac{d g}{d x} = x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x} - x^{2} y^{2} e^{2 x} - y^{2} x e^{2 x}$

Step 12.1.3

Combine the opposite terms in $x y^{2} e^{2 x} + x^{2} y^{2} e^{2 x} + 3 e^{2 x} - x^{2} y^{2} e^{2 x} - y^{2} x e^{2 x}$ .

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Step 12.1.3.1

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

$\frac{d g}{d x} = x y^{2} e^{2 x} + 3 e^{2 x} + 0 - y^{2} x e^{2 x}$

Step 12.1.3.2

Add $x y^{2} e^{2 x} + 3 e^{2 x}$ and $0$ .

$\frac{d g}{d x} = x y^{2} e^{2 x} + 3 e^{2 x} - y^{2} x e^{2 x}$

Step 12.1.3.3

Reorder the factors in the terms $x y^{2} e^{2 x}$ and $- y^{2} x e^{2 x}$ .

$\frac{d g}{d x} = e^{2 x} y^{2} x + 3 e^{2 x} - e^{2 x} y^{2} x$

Step 12.1.3.4

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

$\frac{d g}{d x} = 3 e^{2 x} + 0$

Step 12.1.3.5

Add $3 e^{2 x}$ and $0$ .

$\frac{d g}{d x} = 3 e^{2 x}$

Step 13

Find the antiderivative of $3 e^{2 x}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = 3 e^{2 x}$ .

$\int \frac{d g}{d x} d x = \int 3 e^{2 x} d x$

Step 13.2

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

$g (x) = \int 3 e^{2 x} d x$

Step 13.3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$g (x) = 3 \int e^{2 x} d x$

Step 13.4

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

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Step 13.4.1

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

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Step 13.4.1.1

Differentiate $2 x$ .

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

Step 13.4.1.2

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 13.4.1.3

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

$2 \cdot 1$

Step 13.4.1.4

Multiply $2$ by $1$ .

$2$

Step 13.4.2

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

$g (x) = 3 \int e^{u} \frac{1}{2} d u$

Step 13.5

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

$g (x) = 3 \int \frac{e^{u}}{2} d u$

Step 13.6

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

$g (x) = 3 (\frac{1}{2} \int e^{u} d u)$

Step 13.7

Combine $\frac{1}{2}$ and $3$ .

$g (x) = \frac{3}{2} \int e^{u} d u$

Step 13.8

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

$g (x) = \frac{3}{2} (e^{u} + C)$

Step 13.9

Simplify.

$g (x) = \frac{3}{2} e^{u} + C$

Step 13.10

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

$g (x) = \frac{3}{2} e^{2 x} + C$

Step 14

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

$f (x, y) = \frac{1}{2} x^{2} e^{2 x} y^{2} + \frac{3}{2} e^{2 x} + C$

Step 15

Simplify $f (x, y) = \frac{1}{2} x^{2} e^{2 x} y^{2} + \frac{3}{2} e^{2 x} + C$ .

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Step 15.1

Simplify each term.

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Step 15.1.1

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

$f (x, y) = \frac{x^{2}}{2} e^{2 x} y^{2} + \frac{3}{2} e^{2 x} + C$

Step 15.1.2

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

$f (x, y) = \frac{x^{2} e^{2 x}}{2} y^{2} + \frac{3}{2} e^{2 x} + C$

Step 15.1.3

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

$f (x, y) = \frac{x^{2} e^{2 x} y^{2}}{2} + \frac{3}{2} e^{2 x} + C$

Step 15.1.4

Combine $\frac{3}{2}$ and $e^{2 x}$ .

$f (x, y) = \frac{x^{2} e^{2 x} y^{2}}{2} + \frac{3 e^{2 x}}{2} + C$

Step 15.2

Reorder factors in $f (x, y) = \frac{x^{2} e^{2 x} y^{2}}{2} + \frac{3 e^{2 x}}{2} + C$ .

$f (x, y) = \frac{x^{2} y^{2} e^{2 x}}{2} + \frac{3 e^{2 x}}{2} + C$