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Calculus Examples
Step 1
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, 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 1.2
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
Rewrite as .
Step 1.6.3
Expand using the FOIL Method.
Step 1.6.3.1
Apply the distributive property.
Step 1.6.3.2
Apply the distributive property.
Step 1.6.3.3
Apply the distributive property.
Step 1.6.4
Simplify and combine like terms.
Step 1.6.4.1
Simplify each term.
Step 1.6.4.1.1
Multiply by by adding the exponents.
Step 1.6.4.1.1.1
Use the power rule to combine exponents.
Step 1.6.4.1.1.2
Add and .
Step 1.6.4.1.2
Move to the left of .
Step 1.6.4.1.3
Multiply by .
Step 1.6.4.2
Add and .
Step 1.6.5
Apply the distributive property.
Step 1.6.6
Simplify.
Step 1.6.6.1
Rewrite using the commutative property of multiplication.
Step 1.6.6.2
Move to the left of .
Step 1.6.7
Cancel the common factor of .
Step 1.6.7.1
Cancel the common factor.
Step 1.6.7.2
Divide by .
Step 1.6.8
Apply the distributive property.
Step 1.6.9
Multiply by by adding the exponents.
Step 1.6.9.1
Move .
Step 1.6.9.2
Multiply by .
Step 1.6.10
Cancel the common factor of and .
Step 1.6.10.1
Factor out of .
Step 1.6.10.2
Cancel the common factors.
Step 1.6.10.2.1
Multiply by .
Step 1.6.10.2.2
Cancel the common factor.
Step 1.6.10.2.3
Rewrite the expression.
Step 1.6.10.2.4
Divide by .
Step 1.6.11
Apply the distributive property.
Step 1.6.12
Multiply by by adding the exponents.
Step 1.6.12.1
Multiply by .
Step 1.6.12.1.1
Raise to the power of .
Step 1.6.12.1.2
Use the power rule to combine exponents.
Step 1.6.12.2
Add and .
Step 1.6.13
Move to the left of .
Step 1.6.14
Expand using the FOIL Method.
Step 1.6.14.1
Apply the distributive property.
Step 1.6.14.2
Apply the distributive property.
Step 1.6.14.3
Apply the distributive property.
Step 1.6.15
Simplify each term.
Step 1.6.15.1
Multiply by by adding the exponents.
Step 1.6.15.1.1
Move .
Step 1.6.15.1.2
Multiply by .
Step 1.6.15.1.2.1
Raise to the power of .
Step 1.6.15.1.2.2
Use the power rule to combine exponents.
Step 1.6.15.1.3
Add and .
Step 1.6.15.2
Rewrite using the commutative property of multiplication.
Step 1.6.15.3
Multiply by by adding the exponents.
Step 1.6.15.3.1
Move .
Step 1.6.15.3.2
Multiply by .
Step 1.6.15.4
Rewrite using the commutative property of multiplication.
Step 1.7
Simplify the expression.
Step 1.7.1
Move .
Step 1.7.2
Reorder and .
Step 1.7.3
Reorder and .
Step 1.7.4
Move .
Step 1.7.5
Move .
Step 1.7.6
Move .
Step 1.7.7
Move .
Step 1.7.8
Move .
Step 1.7.9
Move .
Step 2
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 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 2.4
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 2.5
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 2.6
Set up the system of equations to find the coefficients of the partial fractions.
Step 3
Step 3.1
Rewrite the equation as .
Step 3.2
Replace all occurrences of with in each equation.
Step 3.2.1
Replace all occurrences of in with .
Step 3.2.2
Simplify the right side.
Step 3.2.2.1
Simplify .
Step 3.2.2.1.1
Multiply by .
Step 3.2.2.1.2
Add and .
Step 3.2.3
Rewrite the equation as .
Step 3.2.4
Divide each term in by and simplify.
Step 3.2.4.1
Divide each term in by .
Step 3.2.4.2
Simplify the left side.
Step 3.2.4.2.1
Cancel the common factor of .
Step 3.2.4.2.1.1
Cancel the common factor.
Step 3.2.4.2.1.2
Divide by .
Step 3.2.4.3
Simplify the right side.
Step 3.2.4.3.1
Cancel the common factor of and .
Step 3.2.4.3.1.1
Factor out of .
Step 3.2.4.3.1.2
Cancel the common factors.
Step 3.2.4.3.1.2.1
Factor out of .
Step 3.2.4.3.1.2.2
Cancel the common factor.
Step 3.2.4.3.1.2.3
Rewrite the expression.
Step 3.3
Replace all occurrences of with in each equation.
Step 3.3.1
Rewrite the equation as .
Step 3.3.2
Replace all occurrences of in with .
Step 3.3.3
Simplify the right side.
Step 3.3.3.1
Remove parentheses.
Step 3.3.4
Replace all occurrences of in with .
Step 3.3.5
Simplify the right side.
Step 3.3.5.1
Cancel the common factor of .
Step 3.3.5.1.1
Factor out of .
Step 3.3.5.1.2
Cancel the common factor.
Step 3.3.5.1.3
Rewrite the expression.
Step 3.4
Solve for in .
Step 3.4.1
Rewrite the equation as .
Step 3.4.2
Subtract from both sides of the equation.
Step 3.5
Replace all occurrences of with in each equation.
Step 3.5.1
Replace all occurrences of in with .
Step 3.5.2
Simplify the right side.
Step 3.5.2.1
Simplify .
Step 3.5.2.1.1
Cancel the common factor of .
Step 3.5.2.1.1.1
Move the leading negative in into the numerator.
Step 3.5.2.1.1.2
Cancel the common factor.
Step 3.5.2.1.1.3
Rewrite the expression.
Step 3.5.2.1.2
Subtract from .
Step 3.6
Solve for in .
Step 3.6.1
Rewrite the equation as .
Step 3.6.2
Subtract from both sides of the equation.
Step 3.7
Solve the system of equations.
Step 3.8
List all of the solutions.
Step 4
Replace each of the partial fraction coefficients in with the values found for , , , , and .
Step 5
Step 5.1
Remove parentheses.
Step 5.2
Add and .
Step 5.3
Rewrite as .
Step 5.4
Add and .
Step 5.5
Combine and .