Calculus Examples

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

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

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

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Reorder $y$ and $\frac{\frac{2 x + 1}{x + 1}}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{\frac{2 x + 1}{x + 1}}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{\frac{2 x + 1}{x + 1}}{x} d x}$

Step 2.2

Integrate $\frac{\frac{2 x + 1}{x + 1}}{x}$ .

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$e^{\int \frac{2 x + 1}{x + 1} \cdot \frac{1}{x} d x}$

Step 2.2.1.2

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

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

Step 2.2.2

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

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Step 2.2.2.1

Let $u = (x + 1) x$ . Find $\frac{d u}{d x}$ .

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Step 2.2.2.1.1

Differentiate $(x + 1) x$ .

$\frac{d}{d x} [(x + 1) x]$

Step 2.2.2.1.2

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 $f (x) = x + 1$ and $g (x) = x$ .

$(x + 1) \frac{d}{d x} [x] + x \frac{d}{d x} [x + 1]$

Step 2.2.2.1.3

Differentiate.

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Step 2.2.2.1.3.1

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

$(x + 1) \cdot 1 + x \frac{d}{d x} [x + 1]$

Step 2.2.2.1.3.2

Multiply $x + 1$ by $1$ .

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

Step 2.2.2.1.3.3

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

$x + 1 + x (\frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 2.2.2.1.3.4

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

$x + 1 + x (1 + \frac{d}{d x} [1])$

Step 2.2.2.1.3.5

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

$x + 1 + x (1 + 0)$

Step 2.2.2.1.3.6

Simplify by adding terms.

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Step 2.2.2.1.3.6.1

Add $1$ and $0$ .

$x + 1 + x \cdot 1$

Step 2.2.2.1.3.6.2

Multiply $x$ by $1$ .

$x + 1 + x$

Step 2.2.2.1.3.6.3

Add $x$ and $x$ .

$2 x + 1$

Step 2.2.2.2

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

$e^{\int \frac{1}{u} d u}$

Step 2.2.3

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

$e^{\ln (| u |) + C}$

Step 2.2.4

Replace all occurrences of $u$ with $(x + 1) x$ .

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

Step 2.3

Remove the constant of integration.

$e^{\ln ((x + 1) x)}$

Step 2.4

Exponentiation and log are inverse functions.

$(x + 1) x$

Step 2.5

Apply the distributive property.

$x \cdot x + 1 x$

Step 2.6

Multiply $x$ by $x$ .

$x^{2} + 1 x$

Step 2.7

Multiply $x$ by $1$ .

$x^{2} + x$

Step 3

Multiply each term by the integrating factor $x^{2} + x$ .

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Step 3.1

Multiply each term by $x^{2} + x$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

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

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Step 3.2.6.1

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

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

Step 3.2.6.2

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

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

Step 3.2.6.3

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

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

Step 3.2.6.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 3.2.7

Cancel the common factor of $x$ .

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Step 3.2.7.1

Cancel the common factor.

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

Step 3.2.7.2

Rewrite the expression.

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

Step 3.2.8

Cancel the common factor of $x + 1$ .

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Step 3.2.8.1

Cancel the common factor.

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

Step 3.2.8.2

Divide $(2 x + 1) y$ by $1$ .

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

Step 3.2.9

Apply the distributive property.

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

Step 3.2.10

Multiply $y$ by $1$ .

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

Step 3.3

Simplify each term.

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Step 3.3.1

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

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

Step 3.3.2

Move the negative in front of the fraction.

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

Step 3.3.3

Apply the distributive property.

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

Step 3.3.4

Rewrite using the commutative property of multiplication.

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

Step 3.3.5

Rewrite using the commutative property of multiplication.

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

Step 3.3.6

Simplify each term.

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Step 3.3.6.1

Cancel the common factor of $x$ .

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Step 3.3.6.1.1

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

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

Step 3.3.6.1.2

Cancel the common factor.

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

Step 3.3.6.1.3

Rewrite the expression.

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

Step 3.3.6.2

Cancel the common factor of $x$ .

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Step 3.3.6.2.1

Factor $x$ out of $- x$ .

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

Step 3.3.6.2.2

Cancel the common factor.

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

Step 3.3.6.2.3

Rewrite the expression.

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

Step 3.4

Combine the opposite terms in $x^{2} + x - x - 1$ .

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Step 3.4.1

Subtract $x$ from $x$ .

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

Step 3.4.2

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

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

Step 4

Rewrite the left side as a result of differentiating a product.

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 7.3

Apply the constant rule.

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

Step 7.4

Simplify.

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

Step 8

Divide each term in $(x^{2} + x) y = \frac{1}{3} x^{3} - x + C$ by $x^{2} + x$ and simplify.

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Step 8.1

Divide each term in $(x^{2} + x) y = \frac{1}{3} x^{3} - x + C$ by $x^{2} + x$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{3} x^{3}}{x^{2} + x} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$y = \frac{\frac{x^{3}}{3}}{x^{2} + x} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.2

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

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Step 8.3.1.2.1

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

$y = \frac{\frac{x^{3}}{3}}{x \cdot x + x} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.2.2

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

$y = \frac{\frac{x^{3}}{3}}{x \cdot x + x^{1}} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.2.3

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

$y = \frac{\frac{x^{3}}{3}}{x \cdot x + x \cdot 1} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.2.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 8.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x^{3}}{3} \cdot \frac{1}{x (x + 1)} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.4

Combine.

$y = \frac{x^{3} \cdot 1}{3 (x (x + 1))} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.5

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

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Step 8.3.1.5.1

Factor $x$ out of $x^{3} \cdot 1$ .

$y = \frac{x (x^{2} \cdot 1)}{3 (x (x + 1))} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.5.2

Cancel the common factors.

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Step 8.3.1.5.2.1

Factor $x$ out of $3 (x (x + 1))$ .

$y = \frac{x (x^{2} \cdot 1)}{x (3 (x + 1))} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.5.2.2

Cancel the common factor.

$y = \frac{x (x^{2} \cdot 1)}{x (3 (x + 1))} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.5.2.3

Rewrite the expression.

$y = \frac{x^{2} \cdot 1}{3 (x + 1)} + \frac{- x}{x^{2} + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.6

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

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

Step 8.3.1.7

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

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Step 8.3.1.7.1

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

$y = \frac{x^{2}}{3 (x + 1)} + \frac{- x}{x \cdot x + x} + \frac{C}{x^{2} + x}$

Step 8.3.1.7.2

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

$y = \frac{x^{2}}{3 (x + 1)} + \frac{- x}{x \cdot x + x^{1}} + \frac{C}{x^{2} + x}$

Step 8.3.1.7.3

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

$y = \frac{x^{2}}{3 (x + 1)} + \frac{- x}{x \cdot x + x \cdot 1} + \frac{C}{x^{2} + x}$

Step 8.3.1.7.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 8.3.1.8

Cancel the common factor of $x$ .

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Step 8.3.1.8.1

Cancel the common factor.

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

Step 8.3.1.8.2

Rewrite the expression.

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

Step 8.3.1.9

Move the negative in front of the fraction.

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

Step 8.3.1.10

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

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Step 8.3.1.10.1

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

$y = \frac{x^{2}}{3 (x + 1)} - \frac{1}{x + 1} + \frac{C}{x \cdot x + x}$

Step 8.3.1.10.2

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

$y = \frac{x^{2}}{3 (x + 1)} - \frac{1}{x + 1} + \frac{C}{x \cdot x + x^{1}}$

Step 8.3.1.10.3

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

$y = \frac{x^{2}}{3 (x + 1)} - \frac{1}{x + 1} + \frac{C}{x \cdot x + x \cdot 1}$

Step 8.3.1.10.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 8.3.2

To write $- \frac{1}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{x^{2}}{3 (x + 1)} - \frac{1}{x + 1} \cdot \frac{3}{3} + \frac{C}{x (x + 1)}$

Step 8.3.3

Write each expression with a common denominator of $3 (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.3.1

Multiply $\frac{1}{x + 1}$ by $\frac{3}{3}$ .

$y = \frac{x^{2}}{3 (x + 1)} - \frac{3}{(x + 1) \cdot 3} + \frac{C}{x (x + 1)}$

Step 8.3.3.2

Reorder the factors of $(x + 1) \cdot 3$ .

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

Step 8.3.4

Combine the numerators over the common denominator.

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

Step 8.3.5

To write $\frac{x^{2} - 3}{3 (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y = \frac{x^{2} - 3}{3 (x + 1)} \cdot \frac{x}{x} + \frac{C}{x (x + 1)}$

Step 8.3.6

To write $\frac{C}{x (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{x^{2} - 3}{3 (x + 1)} \cdot \frac{x}{x} + \frac{C}{x (x + 1)} \cdot \frac{3}{3}$

Step 8.3.7

Write each expression with a common denominator of $3 (x + 1) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.7.1

Multiply $\frac{x^{2} - 3}{3 (x + 1)}$ by $\frac{x}{x}$ .

$y = \frac{(x^{2} - 3) x}{3 (x + 1) x} + \frac{C}{x (x + 1)} \cdot \frac{3}{3}$

Step 8.3.7.2

Multiply $\frac{C}{x (x + 1)}$ by $\frac{3}{3}$ .

$y = \frac{(x^{2} - 3) x}{3 (x + 1) x} + \frac{C \cdot 3}{x (x + 1) \cdot 3}$

Step 8.3.7.3

Reorder the factors of $3 (x + 1) x$ .

$y = \frac{(x^{2} - 3) x}{3 x (x + 1)} + \frac{C \cdot 3}{x (x + 1) \cdot 3}$

Step 8.3.7.4

Reorder the factors of $x (x + 1) \cdot 3$ .

$y = \frac{(x^{2} - 3) x}{3 x (x + 1)} + \frac{C \cdot 3}{3 x (x + 1)}$

Step 8.3.8

Combine the numerators over the common denominator.

$y = \frac{(x^{2} - 3) x + C \cdot 3}{3 x (x + 1)}$

Step 8.3.9

Simplify the numerator.

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Step 8.3.9.1

Apply the distributive property.

$y = \frac{x^{2} x - 3 x + C \cdot 3}{3 x (x + 1)}$

Step 8.3.9.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.3.9.2.1

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

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Step 8.3.9.2.1.1

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

$y = \frac{x^{2} x^{1} - 3 x + C \cdot 3}{3 x (x + 1)}$

Step 8.3.9.2.1.2

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

$y = \frac{x^{2 + 1} - 3 x + C \cdot 3}{3 x (x + 1)}$

Step 8.3.9.2.2

Add $2$ and $1$ .

$y = \frac{x^{3} - 3 x + C \cdot 3}{3 x (x + 1)}$

Step 8.3.9.3

Move $3$ to the left of $C$ .

$y = \frac{x^{3} - 3 x + 3 C}{3 x (x + 1)}$