Calculus Examples

\frac{x - 3}{x^{3} + 3 x}

$\frac{x - 3}{x^{3} + 3 x}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor

x

$x$ out of

x^{3} + 3 x

$x^{3} + 3 x$ .

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Step 1.1.1

Factor

x

$x$ out of

x^{3}

$x^{3}$ .

\frac{x - 3}{x \cdot x^{2} + 3 x}

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

Step 1.1.2

Factor

x

$x$ out of

3 x

$3 x$ .

\frac{x - 3}{x \cdot x^{2} + x \cdot 3}

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

Step 1.1.3

Factor

x

$x$ out of

x \cdot x^{2} + x \cdot 3

$x \cdot x^{2} + x \cdot 3$ .

\frac{x - 3}{x (x^{2} + 3)}

$\frac{x - 3}{x (x^{2} + 3)}$

\frac{x - 3}{x (x^{2} + 3)}

$\frac{x - 3}{x (x^{2} + 3)}$

Step 1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor is 2nd order,

2

$2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

\frac{A}{x} + \frac{B x + C}{x^{2} + 3}

$\frac{A}{x} + \frac{B x + C}{x^{2} + 3}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is

x (x^{2} + 3)

$x (x^{2} + 3)$ .

\frac{(x - 3) (x (x^{2} + 3))}{x (x^{2} + 3)} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$\frac{(x - 3) (x (x^{2} + 3))}{x (x^{2} + 3)} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

Step 1.4

Cancel the common factor of

x

$x$ .

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Step 1.4.1

Cancel the common factor.

\frac{(x - 3) (x (x^{2} + 3))}{x (x^{2} + 3)} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$\frac{(x - 3) (x (x^{2} + 3))}{x (x^{2} + 3)} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

Step 1.4.2

Rewrite the expression.

\frac{(x - 3) (x^{2} + 3)}{x^{2} + 3} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$\frac{(x - 3) (x^{2} + 3)}{x^{2} + 3} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

\frac{(x - 3) (x^{2} + 3)}{x^{2} + 3} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$\frac{(x - 3) (x^{2} + 3)}{x^{2} + 3} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

Step 1.5

Cancel the common factor of

x^{2} + 3

$x^{2} + 3$ .

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Step 1.5.1

Cancel the common factor.

\frac{(x - 3) (x^{2} + 3)}{x^{2} + 3} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$\frac{(x - 3) (x^{2} + 3)}{x^{2} + 3} = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

Step 1.5.2

Divide

x - 3

$x - 3$ by

1

$1$ .

x - 3 = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$x - 3 = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

x - 3 = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$x - 3 = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of

x

$x$ .

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Step 1.6.1.1

Cancel the common factor.

x - 3 = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$x - 3 = \frac{A (x (x^{2} + 3))}{x} + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

Step 1.6.1.2

Divide

A (x^{2} + 3)

$A (x^{2} + 3)$ by

1

$1$ .

x - 3 = A (x^{2} + 3) + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$x - 3 = A (x^{2} + 3) + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

x - 3 = A (x^{2} + 3) + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$x - 3 = A (x^{2} + 3) + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

Step 1.6.2

Apply the distributive property.

x - 3 = A x^{2} + A \cdot 3 + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

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

Step 1.6.3

Move

3

$3$ to the left of

A

$A$ .

x - 3 = A x^{2} + 3 \cdot A + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

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

Step 1.6.4

Cancel the common factor of

x^{2} + 3

$x^{2} + 3$ .

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Step 1.6.4.1

Cancel the common factor.

x - 3 = A x^{2} + 3 A + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}

$x - 3 = A x^{2} + 3 A + \frac{(B x + C) (x (x^{2} + 3))}{x^{2} + 3}$

Step 1.6.4.2

Divide

(B x + C) (x)

$(B x + C) (x)$ by

1

$1$ .

x - 3 = A x^{2} + 3 A + (B x + C) (x)

$x - 3 = A x^{2} + 3 A + (B x + C) (x)$

x - 3 = A x^{2} + 3 A + (B x + C) (x)

$x - 3 = A x^{2} + 3 A + (B x + C) (x)$

Step 1.6.5

Apply the distributive property.

x - 3 = A x^{2} + 3 A + B x \cdot x + C x

$x - 3 = A x^{2} + 3 A + B x \cdot x + C x$

Step 1.6.6

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 1.6.6.1

Move

x

$x$ .

x - 3 = A x^{2} + 3 A + B (x \cdot x) + C x

$x - 3 = A x^{2} + 3 A + B (x \cdot x) + C x$

Step 1.6.6.2

Multiply

x

$x$ by

x

$x$ .

x - 3 = A x^{2} + 3 A + B x^{2} + C x

$x - 3 = A x^{2} + 3 A + B x^{2} + C x$

x - 3 = A x^{2} + 3 A + B x^{2} + C x

$x - 3 = A x^{2} + 3 A + B x^{2} + C x$

x - 3 = A x^{2} + 3 A + B x^{2} + C x

$x - 3 = A x^{2} + 3 A + B x^{2} + C x$

Step 1.7

Move

3 A

$3 A$ .

x - 3 = A x^{2} + B x^{2} + C x + 3 A

$x - 3 = A x^{2} + B x^{2} + C x + 3 A$

x - 3 = A x^{2} + B x^{2} + C x + 3 A

$x - 3 = A x^{2} + B x^{2} + C x + 3 A$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of

x^{2}

$x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

0 = A + B

$0 = A + B$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of

x

$x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

1 = C

$1 = C$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing

x

$x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

- 3 = 3 A

$- 3 = 3 A$

Step 2.4

Set up the system of equations to find the coefficients of the partial fractions.

0 = A + B

$0 = A + B$

1 = C

$1 = C$

- 3 = 3 A

$- 3 = 3 A$

0 = A + B

$0 = A + B$

1 = C

$1 = C$

- 3 = 3 A

$- 3 = 3 A$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as

C = 1

$C = 1$ .

C = 1

$C = 1$

0 = A + B

$0 = A + B$

- 3 = 3 A

$- 3 = 3 A$

Step 3.2

Replace all occurrences of

C

$C$ with

1

$1$ in each equation.

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Step 3.2.1

Rewrite the equation as

3 A = - 3

$3 A = - 3$ .

3 A = - 3

$3 A = - 3$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

Step 3.2.2

Divide each term in

3 A = - 3

$3 A = - 3$ by

3

$3$ and simplify.

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Step 3.2.2.1

Divide each term in

3 A = - 3

$3 A = - 3$ by

3

$3$ .

\frac{3 A}{3} = \frac{- 3}{3}

$\frac{3 A}{3} = \frac{- 3}{3}$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of

3

$3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

\frac{3 A}{3} = \frac{- 3}{3}

$\frac{3 A}{3} = \frac{- 3}{3}$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

Step 3.2.2.2.1.2

Divide

A

$A$ by

1

$1$ .

A = \frac{- 3}{3}

$A = \frac{- 3}{3}$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

A = \frac{- 3}{3}

$A = \frac{- 3}{3}$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

A = \frac{- 3}{3}

$A = \frac{- 3}{3}$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide

- 3

$- 3$ by

3

$3$ .

A = - 1

$A = - 1$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

A = - 1

$A = - 1$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

A = - 1

$A = - 1$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

A = - 1

$A = - 1$

C = 1

$C = 1$

0 = A + B

$0 = A + B$

Step 3.3

Replace all occurrences of

A

$A$ with

- 1

$- 1$ in each equation.

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Step 3.3.1

Replace all occurrences of

A

$A$ in

0 = A + B

$0 = A + B$ with

- 1

$- 1$ .

0 = (- 1) + B

$0 = (- 1) + B$

A = - 1

$A = - 1$

C = 1

$C = 1$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Remove parentheses.

0 = - 1 + B

$0 = - 1 + B$

A = - 1

$A = - 1$

C = 1

$C = 1$

0 = - 1 + B

$0 = - 1 + B$

A = - 1

$A = - 1$

C = 1

$C = 1$

0 = - 1 + B

$0 = - 1 + B$

A = - 1

$A = - 1$

C = 1

$C = 1$

Step 3.4

Solve for

B

$B$ in

0 = - 1 + B

$0 = - 1 + B$ .

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Step 3.4.1

Rewrite the equation as

- 1 + B = 0

$- 1 + B = 0$ .

- 1 + B = 0

$- 1 + B = 0$

A = - 1

$A = - 1$

C = 1

$C = 1$

Step 3.4.2

Add

1

$1$ to both sides of the equation.

B = 1

$B = 1$

A = - 1

$A = - 1$

C = 1

$C = 1$

B = 1

$B = 1$

A = - 1

$A = - 1$

C = 1

$C = 1$

Step 3.5

Solve the system of equations.

$B = 1 A = - 1 C = 1$

Step 3.6

List all of the solutions.

$B = 1, A = - 1, C = 1$

Step 4

Replace each of the partial fraction coefficients in

$\frac{A}{x} + \frac{B x + C}{x^{2} + 3}$ with the values found for

$A$ ,

$B$ , and

$C$ .

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

Step 5

Simplify.

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Step 5.1

Remove parentheses.

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

Step 5.2

Multiply

$x$ by

$1$ .

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

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