Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Multiply $2$ by $1$ .

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

Step 1.1.1.3

Apply the distributive property.

$2 x y + 2 x + x^{2} \frac{d y}{d x} + 1 \frac{d y}{d x} = 0$

Step 1.1.1.4

Multiply $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

Factor $\frac{d y}{d x}$ out of $x^{2} \frac{d y}{d x}$ .

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

Step 1.1.3.2

Raise $\frac{d y}{d x}$ to the power of $1$ .

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

Step 1.1.3.3

Factor $\frac{d y}{d x}$ out of ${(\frac{d y}{d x})}^{1}$ .

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

Step 1.1.3.4

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

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

Step 1.1.4

Divide each term in $\frac{d y}{d x} (x^{2} + 1) = - 2 x y - 2 x$ by $x^{2} + 1$ and simplify.

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Step 1.1.4.1

Divide each term in $\frac{d y}{d x} (x^{2} + 1) = - 2 x y - 2 x$ by $x^{2} + 1$ .

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

Step 1.1.4.2

Simplify the left side.

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Step 1.1.4.2.1

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

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Step 1.1.4.2.1.1

Cancel the common factor.

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

Step 1.1.4.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.4.3

Simplify the right side.

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Step 1.1.4.3.1

Combine the numerators over the common denominator.

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

Step 1.1.4.3.2

Simplify the numerator.

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Step 1.1.4.3.2.1

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

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Step 1.1.4.3.2.1.1

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

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

Step 1.1.4.3.2.1.2

Factor $2 x$ out of $- 2 x$ .

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

Step 1.1.4.3.2.1.3

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

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

Step 1.1.4.3.2.2

Rewrite $- 1 y$ as $- y$ .

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

Step 1.1.4.3.3

Simplify with factoring out.

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Step 1.1.4.3.3.1

Factor $- 1$ out of $- y$ .

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

Step 1.1.4.3.3.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.1.4.3.3.3

Factor $- 1$ out of $- (y) - 1 (1)$ .

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

Step 1.1.4.3.3.4

Simplify the expression.

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Step 1.1.4.3.3.4.1

Rewrite $- (y + 1)$ as $- 1 (y + 1)$ .

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

Step 1.1.4.3.3.4.2

Move the negative in front of the fraction.

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

Step 1.2

Regroup factors.

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

Step 1.3

Multiply both sides by $\frac{1}{y + 1}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.2

Cancel the common factor of $y + 1$ .

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Step 1.4.2.1

Move the leading negative in $- \frac{1}{y + 1}$ into the numerator.

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

Step 1.4.2.2

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

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

Step 1.4.2.3

Cancel the common factor.

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

Step 1.4.2.4

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u_{1} = y + 1$ . Then $d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.1.1

Let $u_{1} = y + 1$ . Find $\frac{d u_{1}}{d y}$ .

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Step 2.2.1.1.1

Differentiate $y + 1$ .

$\frac{d}{d y} [y + 1]$

Step 2.2.1.1.2

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

$\frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 2.2.1.1.3

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

$1 + \frac{d}{d y} [1]$

Step 2.2.1.1.4

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

$1 + 0$

Step 2.2.1.1.5

Add $1$ and $0$ .

$1$

Step 2.2.1.2

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

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u_{1}$ with $y + 1$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $2$ by $- 1$ .

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

Step 2.3.4

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

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Step 2.3.4.1

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

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Step 2.3.4.1.1

Differentiate $x^{2} + 1$ .

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

Step 2.3.4.1.2

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

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

Step 2.3.4.1.3

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

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

Step 2.3.4.1.4

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

$2 x + 0$

Step 2.3.4.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.4.2

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Move $2$ to the left of $u_{2}$ .

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

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

Step 2.3.7.2

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

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Step 2.3.7.2.1

Factor $2$ out of $- 2$ .

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

Step 2.3.7.2.2

Cancel the common factors.

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Step 2.3.7.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.7.2.2.2

Cancel the common factor.

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

Step 2.3.7.2.2.3

Rewrite the expression.

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

Step 2.3.7.2.2.4

Divide $- 1$ by $1$ .

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

Step 2.3.8

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

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

Step 2.3.9

Simplify.

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

Step 2.3.10

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Expand $(y + 1) (x^{2} + 1)$ using the FOIL Method.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Apply the distributive property.

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

Step 3.4.3

Apply the distributive property.

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

Step 3.5

Simplify each term.

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Step 3.5.1

Multiply $y$ by $1$ .

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

Step 3.5.2

Multiply $x^{2}$ by $1$ .

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

Step 3.5.3

Multiply $1$ by $1$ .

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

Step 3.6

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

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

Step 3.7

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

$e^{C} = | y x^{2} + y + x^{2} + 1 |$

Step 3.8

Solve for $y$ .

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Step 3.8.1

Rewrite the equation as $| y x^{2} + y + x^{2} + 1 | = e^{C}$ .

$| y x^{2} + y + x^{2} + 1 | = e^{C}$

Step 3.8.2

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

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

Step 3.8.3

Move all terms not containing $y$ to the right side of the equation.

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Step 3.8.3.1

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

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

Step 3.8.3.2

Subtract $1$ from both sides of the equation.

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

Step 3.8.4

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

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Step 3.8.4.1

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

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

Step 3.8.4.2

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

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

Step 3.8.4.3

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

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

Step 3.8.4.4

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

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

Step 3.8.5

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

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Step 3.8.5.1

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

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

Step 3.8.5.2

Simplify the left side.

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Step 3.8.5.2.1

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

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Step 3.8.5.2.1.1

Cancel the common factor.

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

Step 3.8.5.2.1.2

Divide $y$ by $1$ .

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

Step 3.8.5.3

Simplify the right side.

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Step 3.8.5.3.1

Simplify each term.

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Step 3.8.5.3.1.1

Move the negative in front of the fraction.

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

Step 3.8.5.3.1.2

Move the negative in front of the fraction.

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

Step 3.8.5.3.2

Combine into one fraction.

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Step 3.8.5.3.2.1

Combine the numerators over the common denominator.

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

Step 3.8.5.3.2.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the constant of integration.

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