Calculus Examples

$(x + 3 x^{3} \sin (y)) d x + (x^{4} \cos (y)) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [3 x^{3} \sin (y)]$

Step 1.3

Evaluate $\frac{d}{d y} [3 x^{3} \sin (y)]$ .

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Step 1.3.1

Since $3 x^{3}$ is constant with respect to $y$ , the derivative of $3 x^{3} \sin (y)$ with respect to $y$ is $3 x^{3} \frac{d}{d y} [\sin (y)]$ .

$\frac{\partial M}{\partial y} = 0 + 3 x^{3} \frac{d}{d y} [\sin (y)]$

Step 1.3.2

The derivative of $\sin (y)$ with respect to $y$ is $\cos (y)$ .

$\frac{\partial M}{\partial y} = 0 + 3 x^{3} \cos (y)$

Step 1.4

Add $0$ and $3 x^{3} \cos (y)$ .

$\frac{\partial M}{\partial y} = 3 x^{3} \cos (y)$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = x^{4} \cos (y)$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{4} \cos (y)]$

Step 2.2

Since $\cos (y)$ is constant with respect to $x$ , the derivative of $x^{4} \cos (y)$ with respect to $x$ is $\cos (y) \frac{d}{d x} [x^{4}]$ .

$\frac{\partial N}{\partial x} = \cos (y) \frac{d}{d x} [x^{4}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{\partial N}{\partial x} = \cos (y) (4 x^{3})$

Step 2.4

Reorder.

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Step 2.4.1

Move $4$ to the left of $\cos (y)$ .

$\frac{\partial N}{\partial x} = 4 \cdot \cos (y) x^{3}$

Step 2.4.2

Reorder the factors of $4 \cos (y) x^{3}$ .

$\frac{\partial N}{\partial x} = 4 x^{3} \cos (y)$

Step 3

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

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Step 3.1

Substitute $3 x^{3} \cos (y)$ for $\frac{\partial M}{\partial y}$ and $4 x^{3} \cos (y)$ for $\frac{\partial N}{\partial x}$ .

$3 x^{3} \cos (y) = 4 x^{3} \cos (y)$

Step 3.2

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

$3 x^{3} \cos (y) = 4 x^{3} \cos (y)$ 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 $3 x^{3} \cos (y)$ for $\frac{\partial M}{\partial y}$ .

$\frac{3 x^{3} \cos (y) - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $4 x^{3} \cos (y)$ for $\frac{\partial N}{\partial x}$ .

$\frac{3 x^{3} \cos (y) - (4 x^{3} \cos (y))}{N}$

Step 4.3

Substitute $x^{4} \cos (y)$ for $N$ .

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Step 4.3.1

Substitute $x^{4} \cos (y)$ for $N$ .

$\frac{3 x^{3} \cos (y) - (4 x^{3} \cos (y))}{x^{4} \cos (y)}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $x^{3} \cos (y)$ out of $3 x^{3} \cos (y) - 1 \cdot 4 x^{3} \cos (y)$ .

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Step 4.3.2.1.1

Factor $x^{3} \cos (y)$ out of $3 x^{3} \cos (y)$ .

$\frac{x^{3} \cos (y) (3) - 1 \cdot 4 x^{3} \cos (y)}{x^{4} \cos (y)}$

Step 4.3.2.1.2

Factor $x^{3} \cos (y)$ out of $- 1 \cdot 4 x^{3} \cos (y)$ .

$\frac{x^{3} \cos (y) (3) + x^{3} \cos (y) (- 1 \cdot 4)}{x^{4} \cos (y)}$

Step 4.3.2.1.3

Factor $x^{3} \cos (y)$ out of $x^{3} \cos (y) (3) + x^{3} \cos (y) (- 1 \cdot 4)$ .

$\frac{x^{3} \cos (y) (3 - 1 \cdot 4)}{x^{4} \cos (y)}$

Step 4.3.2.2

Multiply $- 1$ by $4$ .

$\frac{x^{3} \cos (y) (3 - 4)}{x^{4} \cos (y)}$

Step 4.3.2.3

Subtract $4$ from $3$ .

$\frac{x^{3} \cos (y) \cdot - 1}{x^{4} \cos (y)}$

Step 4.3.3

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 4.3.3.1

Factor $x^{3}$ out of $x^{3} \cos (y) \cdot - 1$ .

$\frac{x^{3} (\cos (y) \cdot - 1)}{x^{4} \cos (y)}$

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $x^{3}$ out of $x^{4} \cos (y)$ .

$\frac{x^{3} (\cos (y) \cdot - 1)}{x^{3} (x \cos (y))}$

Step 4.3.3.2.2

Cancel the common factor.

$\frac{x^{3} (\cos (y) \cdot - 1)}{x^{3} (x \cos (y))}$

Step 4.3.3.2.3

Rewrite the expression.

$\frac{\cos (y) \cdot - 1}{x \cos (y)}$

Step 4.3.4

Cancel the common factor of $\cos (y)$ .

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Step 4.3.4.1

Cancel the common factor.

$\frac{\cos (y) \cdot - 1}{x \cos (y)}$

Step 4.3.4.2

Rewrite the expression.

$\frac{- 1}{x}$

Step 4.3.5

Move the negative in front of the fraction.

$- \frac{1}{x}$

Step 4.4

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

$μ (x, y) = e^{\int - \frac{1}{x} d x}$

Step 5

Evaluate the integral $e^{\int - \frac{1}{x} d x}$ .

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Step 5.1

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

$μ (x, y) = e^{- \int \frac{1}{x} d x}$

Step 5.2

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

$μ (x, y) = e^{- (\ln (| x |) + C)}$

Step 5.3

Simplify.

$μ (x, y) = e^{- \ln (| x |)} + C$

Step 5.4

Simplify each term.

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Step 5.4.1

Simplify $- \ln (| x |)$ by moving $- 1$ inside the logarithm.

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

Step 5.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x |}^{- 1} + C$

Step 5.4.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$μ (x, y) = \frac{1}{| x |} + C$

Step 6

Multiply both sides of $(x + 3 x^{3} \sin (y)) d x + x^{4} \cos (y) d y = 0$ by the integration factor $\frac{1}{x}$ .

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Step 6.1

Multiply $x + 3 x^{3} \sin (y)$ by $\frac{1}{x}$ .

$(x + 3 x^{3} \sin (y)) \frac{1}{x} d x + x^{4} \cos (y) d y = 0$

Step 6.2

Apply the distributive property.

$(x \frac{1}{x} + 3 x^{3} \sin (y) \frac{1}{x}) d x + x^{4} \cos (y) d y = 0$

Step 6.3

Cancel the common factor of $x$ .

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Step 6.3.1

Cancel the common factor.

$(x \frac{1}{x} + 3 x^{3} \sin (y) \frac{1}{x}) d x + x^{4} \cos (y) d y = 0$

Step 6.3.2

Rewrite the expression.

$(1 + 3 x^{3} \sin (y) \frac{1}{x}) d x + x^{4} \cos (y) d y = 0$

Step 6.4

Cancel the common factor of $x$ .

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Step 6.4.1

Factor $x$ out of $3 x^{3} \sin (y)$ .

$(1 + x (3 x^{2} \sin (y)) \frac{1}{x}) d x + x^{4} \cos (y) d y = 0$

Step 6.4.2

Cancel the common factor.

$(1 + x (3 x^{2} \sin (y)) \frac{1}{x}) d x + x^{4} \cos (y) d y = 0$

Step 6.4.3

Rewrite the expression.

$(1 + 3 x^{2} \sin (y)) d x + x^{4} \cos (y) d y = 0$

Step 6.5

Multiply $x^{4} \cos (y)$ by $\frac{1}{x}$ .

$(1 + 3 x^{2} \sin (y)) d x + x^{4} \cos (y) \frac{1}{x} d y = 0$

Step 6.6

Cancel the common factor of $x$ .

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Step 6.6.1

Factor $x$ out of $x^{4} \cos (y)$ .

$(1 + 3 x^{2} \sin (y)) d x + x (x^{3} \cos (y)) \frac{1}{x} d y = 0$

Step 6.6.2

Cancel the common factor.

$(1 + 3 x^{2} \sin (y)) d x + x (x^{3} \cos (y)) \frac{1}{x} d y = 0$

Step 6.6.3

Rewrite the expression.

$(1 + 3 x^{2} \sin (y)) d x + x^{3} \cos (y) d y = 0$

Step 7

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

$f (x, y) = \int x^{3} \cos (y) d y$

Step 8

Integrate $N (x, y) = x^{3} \cos (y)$ to find $f (x, y)$ .

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Step 8.1

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

$f (x, y) = x^{3} \int \cos (y) d y$

Step 8.2

The integral of $\cos (y)$ with respect to $y$ is $\sin (y)$ .

$f (x, y) = x^{3} (\sin (y) + C)$

Step 8.3

Simplify.

$f (x, y) = x^{3} \sin (y) + C$

Step 9

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

$f (x, y) = x^{3} \sin (y) + g (x)$

Step 10

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

$\frac{\partial f}{\partial x} = 1 + 3 x^{2} \sin (y)$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

Evaluate $\frac{d}{d x} [x^{3} \sin (y)]$ .

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Step 11.3.1

Since $\sin (y)$ is constant with respect to $x$ , the derivative of $x^{3} \sin (y)$ with respect to $x$ is $\sin (y) \frac{d}{d x} [x^{3}]$ .

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

Step 11.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 11.3.3

Move $3$ to the left of $\sin (y)$ .

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

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 $3 x^{2} \sin (y)$ from both sides of the equation.

$\frac{d g}{d x} = 1 + 3 x^{2} \sin (y) - 3 x^{2} \sin (y)$

Step 12.1.2

Combine the opposite terms in $1 + 3 x^{2} \sin (y) - 3 x^{2} \sin (y)$ .

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Step 12.1.2.1

Subtract $3 x^{2} \sin (y)$ from $3 x^{2} \sin (y)$ .

$\frac{d g}{d x} = 1 + 0$

Step 12.1.2.2

Add $1$ and $0$ .

$\frac{d g}{d x} = 1$

Step 13

Find the antiderivative of $1$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = 1$ .

$\int \frac{d g}{d x} d x = \int d x$

Step 13.2

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

$g (x) = \int d x$

Step 13.3

Apply the constant rule.

$g (x) = x + C$

Step 14

Substitute for $g (x)$ in $f (x, y) = x^{3} \sin (y) + g (x)$ .

$f (x, y) = x^{3} \sin (y) + x + C$