# Calculus Examples

Step 1
Write the fraction using partial fraction decomposition.
Step 1.1
Decompose the fraction and multiply through by the common denominator.
Step 1.1.1
Factor the fraction.
Step 1.1.1.1
Factor out of .
Step 1.1.1.1.1
Factor out of .
Step 1.1.1.1.2
Factor out of .
Step 1.1.1.1.3
Factor out of .
Step 1.1.1.1.4
Factor out of .
Step 1.1.1.1.5
Factor out of .
Step 1.1.1.2
Factor.
Step 1.1.1.2.1
Factor by grouping.
Step 1.1.1.2.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 1.1.1.2.1.1.1
Factor out of .
Step 1.1.1.2.1.1.2
Rewrite as plus
Step 1.1.1.2.1.1.3
Apply the distributive property.
Step 1.1.1.2.1.2
Factor out the greatest common factor from each group.
Step 1.1.1.2.1.2.1
Group the first two terms and the last two terms.
Step 1.1.1.2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 1.1.1.2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 1.1.1.2.2
Remove unnecessary parentheses.
Step 1.1.2
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 in the denominator is linear, put a single variable in its place .
Step 1.1.3
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 .
Step 1.1.4
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 1.1.5
Cancel the common factor of .
Step 1.1.5.1
Cancel the common factor.
Step 1.1.5.2
Rewrite the expression.
Step 1.1.6
Cancel the common factor of .
Step 1.1.6.1
Cancel the common factor.
Step 1.1.6.2
Rewrite the expression.
Step 1.1.7
Cancel the common factor of .
Step 1.1.7.1
Cancel the common factor.
Step 1.1.7.2
Divide by .
Step 1.1.8
Simplify each term.
Step 1.1.8.1
Cancel the common factor of .
Step 1.1.8.1.1
Cancel the common factor.
Step 1.1.8.1.2
Divide by .
Step 1.1.8.2
Expand using the FOIL Method.
Step 1.1.8.2.1
Apply the distributive property.
Step 1.1.8.2.2
Apply the distributive property.
Step 1.1.8.2.3
Apply the distributive property.
Step 1.1.8.3
Simplify and combine like terms.
Step 1.1.8.3.1
Simplify each term.
Step 1.1.8.3.1.1
Multiply by by adding the exponents.
Step 1.1.8.3.1.1.1
Move .
Step 1.1.8.3.1.1.2
Multiply by .
Step 1.1.8.3.1.2
Multiply by .
Step 1.1.8.3.1.3
Rewrite as .
Step 1.1.8.3.1.4
Multiply by .
Step 1.1.8.3.2
Subtract from .
Step 1.1.8.4
Apply the distributive property.
Step 1.1.8.5
Simplify.
Step 1.1.8.5.1
Rewrite using the commutative property of multiplication.
Step 1.1.8.5.2
Rewrite using the commutative property of multiplication.
Step 1.1.8.5.3
Move to the left of .
Step 1.1.8.6
Cancel the common factor of .
Step 1.1.8.6.1
Cancel the common factor.
Step 1.1.8.6.2
Divide by .
Step 1.1.8.7
Apply the distributive property.
Step 1.1.8.8
Multiply by .
Step 1.1.8.9
Move to the left of .
Step 1.1.8.10
Apply the distributive property.
Step 1.1.8.11
Rewrite using the commutative property of multiplication.
Step 1.1.8.12
Cancel the common factor of .
Step 1.1.8.12.1
Cancel the common factor.
Step 1.1.8.12.2
Divide by .
Step 1.1.8.13
Apply the distributive property.
Step 1.1.8.14
Rewrite using the commutative property of multiplication.
Step 1.1.8.15
Move to the left of .
Step 1.1.8.16
Simplify each term.
Step 1.1.8.16.1
Multiply by by adding the exponents.
Step 1.1.8.16.1.1
Move .
Step 1.1.8.16.1.2
Multiply by .
Step 1.1.8.16.2
Rewrite as .
Step 1.1.8.17
Apply the distributive property.
Step 1.1.8.18
Rewrite using the commutative property of multiplication.
Step 1.1.8.19
Rewrite using the commutative property of multiplication.
Step 1.1.9
Simplify the expression.
Step 1.1.9.1
Move .
Step 1.1.9.2
Move .
Step 1.1.9.3
Reorder and .
Step 1.1.9.4
Move .
Step 1.1.9.5
Move .
Step 1.1.9.6
Move .
Step 1.1.9.7
Move .
Step 1.2
Create equations for the partial fraction variables and use them to set up a system of equations.
Step 1.2.1
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 1.2.2
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 1.2.3
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 1.2.4
Set up the system of equations to find the coefficients of the partial fractions.
Step 1.3
Solve the system of equations.
Step 1.3.1
Solve for in .
Step 1.3.1.1
Rewrite the equation as .
Step 1.3.1.2
Divide each term in by and simplify.
Step 1.3.1.2.1
Divide each term in by .
Step 1.3.1.2.2
Simplify the left side.
Step 1.3.1.2.2.1
Cancel the common factor of .
Step 1.3.1.2.2.1.1
Cancel the common factor.
Step 1.3.1.2.2.1.2
Divide by .
Step 1.3.1.2.3
Simplify the right side.
Step 1.3.1.2.3.1
Dividing two negative values results in a positive value.
Step 1.3.2
Replace all occurrences of with in each equation.
Step 1.3.2.1
Replace all occurrences of in with .
Step 1.3.2.2
Simplify the right side.
Step 1.3.2.2.1
Cancel the common factor of .
Step 1.3.2.2.1.1
Cancel the common factor.
Step 1.3.2.2.1.2
Rewrite the expression.
Step 1.3.2.3
Replace all occurrences of in with .
Step 1.3.2.4
Simplify the right side.
Step 1.3.2.4.1
Simplify each term.
Step 1.3.2.4.1.1
Combine and .
Step 1.3.2.4.1.2
Rewrite as .
Step 1.3.3
Solve for in .
Step 1.3.3.1
Rewrite the equation as .
Step 1.3.3.2
Move all terms not containing to the right side of the equation.
Step 1.3.3.2.1
Subtract from both sides of the equation.
Step 1.3.3.2.2
Subtract from both sides of the equation.
Step 1.3.3.2.3
Subtract from .
Step 1.3.3.2.4
Subtract from .
Step 1.3.4
Replace all occurrences of with in each equation.
Step 1.3.4.1
Replace all occurrences of in with .
Step 1.3.4.2
Simplify the right side.
Step 1.3.4.2.1
Simplify .
Step 1.3.4.2.1.1
Multiply by .
Step 1.3.4.2.1.2
Subtract from .
Step 1.3.5
Solve for in .
Step 1.3.5.1
Rewrite the equation as .
Step 1.3.5.2
Move all terms not containing to the right side of the equation.
Step 1.3.5.2.1
Subtract from both sides of the equation.
Step 1.3.5.2.2
To write as a fraction with a common denominator, multiply by .
Step 1.3.5.2.3
Combine and .
Step 1.3.5.2.4
Combine the numerators over the common denominator.
Step 1.3.5.2.5
Simplify the numerator.
Step 1.3.5.2.5.1
Multiply by .
Step 1.3.5.2.5.2
Subtract from .
Step 1.3.5.3
Divide each term in by and simplify.
Step 1.3.5.3.1
Divide each term in by .
Step 1.3.5.3.2
Simplify the left side.
Step 1.3.5.3.2.1
Cancel the common factor of .
Step 1.3.5.3.2.1.1
Cancel the common factor.
Step 1.3.5.3.2.1.2
Divide by .
Step 1.3.5.3.3
Simplify the right side.
Step 1.3.5.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.3.5.3.3.2
Move the negative in front of the fraction.
Step 1.3.5.3.3.3
Multiply .
Step 1.3.5.3.3.3.1
Multiply by .
Step 1.3.5.3.3.3.2
Multiply by .
Step 1.3.6
Replace all occurrences of with in each equation.
Step 1.3.6.1
Replace all occurrences of in with .
Step 1.3.6.2
Simplify the right side.
Step 1.3.6.2.1
Simplify .
Step 1.3.6.2.1.1
Cancel the common factor of .
Step 1.3.6.2.1.1.1
Move the leading negative in into the numerator.
Step 1.3.6.2.1.1.2
Factor out of .
Step 1.3.6.2.1.1.3
Factor out of .
Step 1.3.6.2.1.1.4
Cancel the common factor.
Step 1.3.6.2.1.1.5
Rewrite the expression.
Step 1.3.6.2.1.2
Move the negative in front of the fraction.
Step 1.3.6.2.1.3
Multiply .
Step 1.3.6.2.1.3.1
Multiply by .
Step 1.3.6.2.1.3.2
Multiply by .
Step 1.3.7
List all of the solutions.
Step 1.4
Replace each of the partial fraction coefficients in with the values found for , , and .
Step 1.5
Simplify.
Step 1.5.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.2
Multiply by .
Step 1.5.3
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.4
Multiply by .
Step 1.5.5
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.6
Multiply by .
Step 1.5.7
Move to the left of .
Step 2
Split the single integral into multiple integrals.
Step 3
Since is constant with respect to , move out of the integral.
Step 4
The integral of with respect to is .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Let . Then , so . Rewrite using and .
Step 6.1
Let . Find .
Step 6.1.1
Differentiate .
Step 6.1.2
By the Sum Rule, the derivative of with respect to is .
Step 6.1.3
Evaluate .
Step 6.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.1.3.2
Differentiate using the Power Rule which states that is where .
Step 6.1.3.3
Multiply by .
Step 6.1.4
Differentiate using the Constant Rule.
Step 6.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.1.4.2
Step 6.2
Rewrite the problem using and .
Step 7
Simplify.
Step 7.1
Multiply by .
Step 7.2
Move to the left of .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
Simplify.
Step 9.1
Multiply by .
Step 9.2
Multiply by .
Step 10
The integral of with respect to is .
Step 11
Since is constant with respect to , move out of the integral.
Step 12
Since is constant with respect to , move out of the integral.
Step 13
Let . Then . Rewrite using and .
Step 13.1
Let . Find .
Step 13.1.1
Differentiate .
Step 13.1.2
By the Sum Rule, the derivative of with respect to is .
Step 13.1.3
Differentiate using the Power Rule which states that is where .
Step 13.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 13.1.5
Step 13.2
Rewrite the problem using and .
Step 14
The integral of with respect to is .
Step 15
Simplify.
Step 16
Substitute back in for each integration substitution variable.
Step 16.1
Replace all occurrences of with .
Step 16.2
Replace all occurrences of with .