Calculus Examples

$2 x (y d) x + (1 + x^{2}) d y = 0$

Step 1

Subtract $2 x y d x$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

Factor $1 + x^{2}$ out of $(1 + x^{2}) y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Combine $- 2$ and $\frac{1}{(1 + x^{2}) y}$ .

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

Step 3.4

Cancel the common factor of $y$ .

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Step 3.4.1

Factor $y$ out of $(1 + x^{2}) y$ .

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

Step 3.4.2

Factor $y$ out of $x y$ .

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 3.5

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

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

$\ln (| y |) + C_{1} = \int - \frac{2 x}{1 + x^{2}} d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

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

$\ln (| y |) + C_{1} = - \int \frac{2 x}{1 + x^{2}} d x$

Step 4.3.2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\ln (| y |) + C_{1} = - (2 \int \frac{x}{1 + x^{2}} d x)$

Step 4.3.3

Multiply $2$ by $- 1$ .

$\ln (| y |) + C_{1} = - 2 \int \frac{x}{1 + x^{2}} d x$

Step 4.3.4

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

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Step 4.3.4.1

Let $u = 1 + x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 4.3.4.1.1

Differentiate $1 + x^{2}$ .

$\frac{d}{d x} [1 + x^{2}]$

Step 4.3.4.1.2

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

$\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$

Step 4.3.4.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 4.3.4.1.4

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

$0 + 2 x$

Step 4.3.4.1.5

Add $0$ and $2 x$ .

$2 x$

Step 4.3.4.2

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

$\ln (| y |) + C_{1} = - 2 \int \frac{1}{u} \cdot \frac{1}{2} d u$

Step 4.3.5

Simplify.

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Step 4.3.5.1

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

$\ln (| y |) + C_{1} = - 2 \int \frac{1}{u \cdot 2} d u$

Step 4.3.5.2

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

$\ln (| y |) + C_{1} = - 2 \int \frac{1}{2 u} d u$

Step 4.3.6

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

$\ln (| y |) + C_{1} = - 2 (\frac{1}{2} \int \frac{1}{u} d u)$

Step 4.3.7

Simplify.

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Step 4.3.7.1

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

$\ln (| y |) + C_{1} = \frac{- 2}{2} \int \frac{1}{u} d u$

Step 4.3.7.2

Cancel the common factor of $- 2$ and $2$ .

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Step 4.3.7.2.1

Factor $2$ out of $- 2$ .

$\ln (| y |) + C_{1} = \frac{2 \cdot - 1}{2} \int \frac{1}{u} d u$

Step 4.3.7.2.2

Cancel the common factors.

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Step 4.3.7.2.2.1

Factor $2$ out of $2$ .

$\ln (| y |) + C_{1} = \frac{2 \cdot - 1}{2 (1)} \int \frac{1}{u} d u$

Step 4.3.7.2.2.2

Cancel the common factor.

$\ln (| y |) + C_{1} = \frac{2 \cdot - 1}{2 \cdot 1} \int \frac{1}{u} d u$

Step 4.3.7.2.2.3

Rewrite the expression.

$\ln (| y |) + C_{1} = \frac{- 1}{1} \int \frac{1}{u} d u$

Step 4.3.7.2.2.4

Divide $- 1$ by $1$ .

$\ln (| y |) + C_{1} = - \int \frac{1}{u} d u$

Step 4.3.8

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

$\ln (| y |) + C_{1} = - (\ln (| u |) + C_{2})$

Step 4.3.9

Simplify.

$\ln (| y |) + C_{1} = - \ln (| u |) + C_{2}$

Step 4.3.10

Replace all occurrences of $u$ with $1 + x^{2}$ .

$\ln (| y |) + C_{1} = - \ln (| 1 + x^{2} |) + C_{2}$

Step 4.4

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

$\ln (| y |) = - \ln (| 1 + x^{2} |) + C$

Step 5

Solve for $y$ .

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Step 5.1

Move all the terms containing a logarithm to the left side of the equation.

$\ln (| y |) + \ln (| 1 + x^{2} |) = C$

Step 5.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (| y | | 1 + x^{2} |) = C$

Step 5.3

To multiply absolute values, multiply the terms inside each absolute value.

$\ln (| y (1 + x^{2}) |) = C$

Step 5.4

Apply the distributive property.

$\ln (| y \cdot 1 + y x^{2} |) = C$

Step 5.5

Multiply $y$ by $1$ .

$\ln (| y + y x^{2} |) = C$

Step 5.6

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y + y x^{2} |)} = e^{C}$

Step 5.7

Rewrite $\ln (| y + y x^{2} |) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = | y + y x^{2} |$

Step 5.8

Solve for $y$ .

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Step 5.8.1

Rewrite the equation as $| y + y x^{2} | = e^{C}$ .

$| y + y x^{2} | = e^{C}$

Step 5.8.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y + y x^{2} = \pm e^{C}$

Step 5.8.3

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

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Step 5.8.3.1

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

$y \cdot 1 + y x^{2} = \pm e^{C}$

Step 5.8.3.2

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

$y \cdot 1 + y (x^{2}) = \pm e^{C}$

Step 5.8.3.3

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

$y (1 + x^{2}) = \pm e^{C}$

Step 5.8.4

Divide each term in $y (1 + x^{2}) = \pm e^{C}$ by $1 + x^{2}$ and simplify.

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Step 5.8.4.1

Divide each term in $y (1 + x^{2}) = \pm e^{C}$ by $1 + x^{2}$ .

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

Step 5.8.4.2

Simplify the left side.

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Step 5.8.4.2.1

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

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Step 5.8.4.2.1.1

Cancel the common factor.

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

Step 5.8.4.2.1.2

Divide $y$ by $1$ .

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

Step 6

Simplify the constant of integration.

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