# Calculus Examples

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 in the denominator is linear, put a single variable in its place .

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 in the denominator is linear, put a single variable in its place .

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

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Rewrite the expression.

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

Simplify each term.

Multiply by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

Apply the distributive property.

Move to the left of the expression .

Multiply by .

Multiply by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

Apply the distributive property.

Move to the left of the expression .

Multiply by .

Remove unnecessary parentheses.

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

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

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

Solve for in the first equation.

Replace all occurrences of with the solution found by solving the last equation for . In this case, the value substituted is .

Solve for in the second equation.

Replace all occurrences of with the solution found by solving the last equation for . In this case, the value substituted is .

Simplify the right side.

Multiply by .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Multiply by .

Add and .

List the solutions to the system of equations.

Replace each of the partial fraction coefficients in with the values found for , and .

Multiply the numerator by the reciprocal of the denominator.

Multiply and .

Multiply the numerator by the reciprocal of the denominator.

Multiply and .