# Calculus Examples

Split Using Partial Fraction Decomposition
Step 1
Decompose the fraction and multiply through by the common denominator.
Step 1.1
Factor out of .
Step 1.1.1
Factor out of .
Step 1.1.2
Factor out of .
Step 1.1.3
Factor out of .
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, 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.
Step 1.3
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 1.4
Cancel the common factor of .
Step 1.4.1
Cancel the common factor.
Step 1.4.2
Rewrite the expression.
Step 1.5
Cancel the common factor of .
Step 1.5.1
Cancel the common factor.
Step 1.5.2
Divide by .
Step 1.6
Simplify each term.
Step 1.6.1
Cancel the common factor of .
Step 1.6.1.1
Cancel the common factor.
Step 1.6.1.2
Divide by .
Step 1.6.2
Apply the distributive property.
Step 1.6.3
Move to the left of .
Step 1.6.4
Cancel the common factor of .
Step 1.6.4.1
Cancel the common factor.
Step 1.6.4.2
Divide by .
Step 1.6.5
Apply the distributive property.
Step 1.6.6
Multiply by by adding the exponents.
Step 1.6.6.1
Move .
Step 1.6.6.2
Multiply by .
Step 1.7
Move .
Step 2
Create equations for the partial fraction variables and use them to set up a system of equations.
Step 2.1
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.
Step 2.2
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.
Step 2.3
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.
Step 2.4
Set up the system of equations to find the coefficients of the partial fractions.
Step 3
Solve the system of equations.
Step 3.1
Rewrite the equation as .
Step 3.2
Replace all occurrences of with in each equation.
Step 3.2.1
Rewrite the equation as .
Step 3.2.2
Divide each term in by and simplify.
Step 3.2.2.1
Divide each term in by .
Step 3.2.2.2
Simplify the left side.
Step 3.2.2.2.1
Cancel the common factor of .
Step 3.2.2.2.1.1
Cancel the common factor.
Step 3.2.2.2.1.2
Divide by .
Step 3.2.2.3
Simplify the right side.
Step 3.2.2.3.1
Divide by .
Step 3.3
Replace all occurrences of with in each equation.
Step 3.3.1
Replace all occurrences of in with .
Step 3.3.2
Simplify the right side.
Step 3.3.2.1
Remove parentheses.
Step 3.4
Solve for in .
Step 3.4.1
Rewrite the equation as .
Step 3.4.2
Add to both sides of the equation.
Step 3.5
Solve the system of equations.
Step 3.6
List all of the solutions.
Step 4
Replace each of the partial fraction coefficients in with the values found for , , and .
Step 5
Simplify.
Step 5.1
Remove parentheses.
Step 5.2
Multiply by .