Calculus Examples

$\frac{- 4 s^{2} + s + 1}{s (s^{2} + 1)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

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, $2$ 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.

$\frac{A}{s} + \frac{B s + C}{s^{2} + 1}$

Step 1.2

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $s (s^{2} + 1)$ .

$\frac{(- 4 s^{2} + s + 1) (s (s^{2} + 1))}{s (s^{2} + 1)} = \frac{A (s (s^{2} + 1))}{s} + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.3

Cancel the common factor of $s$ .

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Step 1.3.1

Cancel the common factor.

$\frac{(- 4 s^{2} + s + 1) (s (s^{2} + 1))}{s (s^{2} + 1)} = \frac{A (s (s^{2} + 1))}{s} + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.3.2

Rewrite the expression.

$\frac{(- 4 s^{2} + s + 1) (s^{2} + 1)}{s^{2} + 1} = \frac{A (s (s^{2} + 1))}{s} + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.4

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

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Step 1.4.1

Cancel the common factor.

$\frac{(- 4 s^{2} + s + 1) (s^{2} + 1)}{s^{2} + 1} = \frac{A (s (s^{2} + 1))}{s} + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.4.2

Divide $- 4 s^{2} + s + 1$ by $1$ .

$- 4 s^{2} + s + 1 = \frac{A (s (s^{2} + 1))}{s} + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.5

Simplify each term.

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Step 1.5.1

Cancel the common factor of $s$ .

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Step 1.5.1.1

Cancel the common factor.

$- 4 s^{2} + s + 1 = \frac{A (s (s^{2} + 1))}{s} + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.5.1.2

Divide $A (s^{2} + 1)$ by $1$ .

$- 4 s^{2} + s + 1 = A (s^{2} + 1) + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.5.2

Apply the distributive property.

$- 4 s^{2} + s + 1 = A s^{2} + A \cdot 1 + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.5.3

Multiply $A$ by $1$ .

$- 4 s^{2} + s + 1 = A s^{2} + A + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.5.4

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

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Step 1.5.4.1

Cancel the common factor.

$- 4 s^{2} + s + 1 = A s^{2} + A + \frac{(B s + C) (s (s^{2} + 1))}{s^{2} + 1}$

Step 1.5.4.2

Divide $(B s + C) (s)$ by $1$ .

$- 4 s^{2} + s + 1 = A s^{2} + A + (B s + C) (s)$

Step 1.5.5

Apply the distributive property.

$- 4 s^{2} + s + 1 = A s^{2} + A + B s \cdot s + C s$

Step 1.5.6

Multiply $s$ by $s$ by adding the exponents.

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Step 1.5.6.1

Move $s$ .

$- 4 s^{2} + s + 1 = A s^{2} + A + B (s \cdot s) + C s$

Step 1.5.6.2

Multiply $s$ by $s$ .

$- 4 s^{2} + s + 1 = A s^{2} + A + B s^{2} + C s$

Step 1.6

Move $A$ .

$- 4 s^{2} + s + 1 = A s^{2} + B s^{2} + C s + A$

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

$- 4 = A + B$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of $s$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = C$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $s$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = A$

Step 2.4

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

$- 4 = A + B$

$1 = C$

$1 = A$

$- 4 = A + B$

$1 = C$

$1 = A$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $C = 1$ .

$C = 1$

$- 4 = A + B$

$1 = A$

Step 3.2

Rewrite the equation as $A = 1$ .

$A = 1$

$C = 1$

$- 4 = A + B$

Step 3.3

Replace all occurrences of $A$ with $1$ in each equation.

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Step 3.3.1

Replace all occurrences of $A$ in $- 4 = A + B$ with $1$ .

$- 4 = (1) + B$

$A = 1$

$C = 1$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Remove parentheses.

$- 4 = 1 + B$

$A = 1$

$C = 1$

$- 4 = 1 + B$

$A = 1$

$C = 1$

$- 4 = 1 + B$

$A = 1$

$C = 1$

Step 3.4

Solve for $B$ in $- 4 = 1 + B$ .

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Step 3.4.1

Rewrite the equation as $1 + B = - 4$ .

$1 + B = - 4$

$A = 1$

$C = 1$

Step 3.4.2

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

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Step 3.4.2.1

Subtract $1$ from both sides of the equation.

$B = - 4 - 1$

$A = 1$

$C = 1$

Step 3.4.2.2

Subtract $1$ from $- 4$ .

$B = - 5$

$A = 1$

$C = 1$

$B = - 5$

$A = 1$

$C = 1$

$B = - 5$

$A = 1$

$C = 1$

Step 3.5

Solve the system of equations.

$B = - 5 A = 1 C = 1$

Step 3.6

List all of the solutions.

$B = - 5, A = 1, C = 1$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{s} + \frac{B s + C}{s^{2} + 1}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{1}{s} + \frac{- 5 s + 1}{s^{2} + 1}$