Calculus Examples

$\frac{1}{x^{2}}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $x^{2}$ .

$\frac{1 (x^{2})}{x^{2}} = \frac{A (x^{2})}{x^{2}} + \frac{B (x^{2})}{x}$

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

$\frac{1 x^{2}}{x^{2}} = \frac{A (x^{2})}{x^{2}} + \frac{B (x^{2})}{x}$

Step 1.2.2

Rewrite the expression.

$1 = \frac{A (x^{2})}{x^{2}} + \frac{B (x^{2})}{x}$

Step 1.3

Simplify each term.

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Step 1.3.1

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

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Step 1.3.1.1

Cancel the common factor.

$1 = \frac{A x^{2}}{x^{2}} + \frac{B (x^{2})}{x}$

Step 1.3.1.2

Divide $A$ by $1$ .

$1 = A + \frac{B (x^{2})}{x}$

Step 1.3.2

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

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Step 1.3.2.1

Factor $x$ out of $B (x^{2})$ .

$1 = A + \frac{x (B (x))}{x}$

Step 1.3.2.2

Cancel the common factors.

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Step 1.3.2.2.1

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

$1 = A + \frac{x (B (x))}{x}$

Step 1.3.2.2.2

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

$1 = A + \frac{x (B (x))}{x \cdot 1}$

Step 1.3.2.2.3

Cancel the common factor.

$1 = A + \frac{x (B (x))}{x \cdot 1}$

Step 1.3.2.2.4

Rewrite the expression.

$1 = A + \frac{B (x)}{1}$

Step 1.3.2.2.5

Divide $B (x)$ by $1$ .

$1 = A + B (x)$

$1 = A + B x$

Step 1.4

Reorder $A$ and $B x$ .

$1 = B x + A$

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

$0 = B$

Step 2.2

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

$1 = A$

Step 2.3

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

$0 = B$

$1 = A$

$0 = B$

$1 = A$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $B = 0$ .

$B = 0$

$1 = A$

Step 3.2

Rewrite the equation as $A = 1$ .

$A = 1$

$B = 0$

Step 3.3

List all of the solutions.

$A = 1, B = 0$

Step 4

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

$\frac{1}{x^{2}} + \frac{0}{x}$