Enter a problem...
Calculus Examples
Step 1
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
Rewrite as .
Step 1.1.1.2
Rewrite as .
Step 1.1.1.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.1.1.4
Simplify.
Step 1.1.1.4.1
Rewrite as .
Step 1.1.1.4.2
Rewrite as .
Step 1.1.1.4.3
Factor.
Step 1.1.1.4.3.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.1.1.4.3.2
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 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 1.1.3
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
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 1.1.5
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
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
Rewrite the expression.
Step 1.1.8
Cancel the common factor of .
Step 1.1.8.1
Cancel the common factor.
Step 1.1.8.2
Divide by .
Step 1.1.9
Simplify each term.
Step 1.1.9.1
Cancel the common factor of .
Step 1.1.9.1.1
Cancel the common factor.
Step 1.1.9.1.2
Divide by .
Step 1.1.9.2
Expand using the FOIL Method.
Step 1.1.9.2.1
Apply the distributive property.
Step 1.1.9.2.2
Apply the distributive property.
Step 1.1.9.2.3
Apply the distributive property.
Step 1.1.9.3
Simplify each term.
Step 1.1.9.3.1
Rewrite using the commutative property of multiplication.
Step 1.1.9.3.2
Multiply by by adding the exponents.
Step 1.1.9.3.2.1
Move .
Step 1.1.9.3.2.2
Multiply by .
Step 1.1.9.3.3
Multiply by .
Step 1.1.9.3.4
Rewrite using the commutative property of multiplication.
Step 1.1.9.3.5
Multiply by .
Step 1.1.9.4
Expand by multiplying each term in the first expression by each term in the second expression.
Step 1.1.9.5
Simplify each term.
Step 1.1.9.5.1
Multiply by by adding the exponents.
Step 1.1.9.5.1.1
Move .
Step 1.1.9.5.1.2
Multiply by .
Step 1.1.9.5.1.2.1
Raise to the power of .
Step 1.1.9.5.1.2.2
Use the power rule to combine exponents.
Step 1.1.9.5.1.3
Add and .
Step 1.1.9.5.2
Multiply by .
Step 1.1.9.5.3
Multiply by .
Step 1.1.9.5.4
Rewrite using the commutative property of multiplication.
Step 1.1.9.5.5
Multiply by by adding the exponents.
Step 1.1.9.5.5.1
Move .
Step 1.1.9.5.5.2
Multiply by .
Step 1.1.9.5.6
Move to the left of .
Step 1.1.9.5.7
Rewrite as .
Step 1.1.9.5.8
Multiply by by adding the exponents.
Step 1.1.9.5.8.1
Move .
Step 1.1.9.5.8.2
Multiply by .
Step 1.1.9.5.9
Multiply by .
Step 1.1.9.5.10
Multiply by .
Step 1.1.9.5.11
Rewrite using the commutative property of multiplication.
Step 1.1.9.5.12
Move to the left of .
Step 1.1.9.5.13
Rewrite as .
Step 1.1.9.6
Combine the opposite terms in .
Step 1.1.9.6.1
Add and .
Step 1.1.9.6.2
Add and .
Step 1.1.9.6.3
Add and .
Step 1.1.9.6.4
Add and .
Step 1.1.9.7
Cancel the common factor of .
Step 1.1.9.7.1
Cancel the common factor.
Step 1.1.9.7.2
Divide by .
Step 1.1.9.8
Apply the distributive property.
Step 1.1.9.9
Rewrite using the commutative property of multiplication.
Step 1.1.9.10
Multiply by .
Step 1.1.9.11
Expand using the FOIL Method.
Step 1.1.9.11.1
Apply the distributive property.
Step 1.1.9.11.2
Apply the distributive property.
Step 1.1.9.11.3
Apply the distributive property.
Step 1.1.9.12
Simplify each term.
Step 1.1.9.12.1
Multiply by by adding the exponents.
Step 1.1.9.12.1.1
Move .
Step 1.1.9.12.1.2
Multiply by .
Step 1.1.9.12.1.2.1
Raise to the power of .
Step 1.1.9.12.1.2.2
Use the power rule to combine exponents.
Step 1.1.9.12.1.3
Add and .
Step 1.1.9.12.2
Multiply by .
Step 1.1.9.12.3
Multiply by .
Step 1.1.9.12.4
Rewrite using the commutative property of multiplication.
Step 1.1.9.12.5
Move to the left of .
Step 1.1.9.12.6
Rewrite as .
Step 1.1.9.13
Cancel the common factor of .
Step 1.1.9.13.1
Cancel the common factor.
Step 1.1.9.13.2
Divide by .
Step 1.1.9.14
Apply the distributive property.
Step 1.1.9.15
Rewrite using the commutative property of multiplication.
Step 1.1.9.16
Multiply by .
Step 1.1.9.17
Expand using the FOIL Method.
Step 1.1.9.17.1
Apply the distributive property.
Step 1.1.9.17.2
Apply the distributive property.
Step 1.1.9.17.3
Apply the distributive property.
Step 1.1.9.18
Simplify each term.
Step 1.1.9.18.1
Multiply by by adding the exponents.
Step 1.1.9.18.1.1
Move .
Step 1.1.9.18.1.2
Multiply by .
Step 1.1.9.18.1.2.1
Raise to the power of .
Step 1.1.9.18.1.2.2
Use the power rule to combine exponents.
Step 1.1.9.18.1.3
Add and .
Step 1.1.9.18.2
Multiply by .
Step 1.1.9.18.3
Multiply by .
Step 1.1.9.18.4
Rewrite using the commutative property of multiplication.
Step 1.1.9.18.5
Multiply by .
Step 1.1.10
Reorder.
Step 1.1.10.1
Move .
Step 1.1.10.2
Move .
Step 1.1.10.3
Move .
Step 1.1.10.4
Move .
Step 1.1.10.5
Move .
Step 1.1.10.6
Move .
Step 1.1.10.7
Move .
Step 1.1.10.8
Move .
Step 1.1.10.9
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 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 1.2.4
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 1.2.5
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
Move all terms not containing to the right side of the equation.
Step 1.3.1.2.1
Subtract from both sides of the equation.
Step 1.3.1.2.2
Subtract from both sides of the equation.
Step 1.3.1.3
Divide each term in by and simplify.
Step 1.3.1.3.1
Divide each term in by .
Step 1.3.1.3.2
Simplify the left side.
Step 1.3.1.3.2.1
Cancel the common factor of .
Step 1.3.1.3.2.1.1
Cancel the common factor.
Step 1.3.1.3.2.1.2
Divide by .
Step 1.3.1.3.3
Simplify the right side.
Step 1.3.1.3.3.1
Simplify each term.
Step 1.3.1.3.3.1.1
Cancel the common factor of and .
Step 1.3.1.3.3.1.1.1
Factor out of .
Step 1.3.1.3.3.1.1.2
Cancel the common factors.
Step 1.3.1.3.3.1.1.2.1
Factor out of .
Step 1.3.1.3.3.1.1.2.2
Cancel the common factor.
Step 1.3.1.3.3.1.1.2.3
Rewrite the expression.
Step 1.3.1.3.3.1.1.2.4
Divide by .
Step 1.3.1.3.3.1.2
Cancel the common factor of and .
Step 1.3.1.3.3.1.2.1
Factor out of .
Step 1.3.1.3.3.1.2.2
Cancel the common factors.
Step 1.3.1.3.3.1.2.2.1
Factor out of .
Step 1.3.1.3.3.1.2.2.2
Cancel the common factor.
Step 1.3.1.3.3.1.2.2.3
Rewrite the expression.
Step 1.3.1.3.3.1.2.2.4
Divide by .
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
Simplify .
Step 1.3.2.2.1.1
Simplify each term.
Step 1.3.2.2.1.1.1
Apply the distributive property.
Step 1.3.2.2.1.1.2
Multiply by .
Step 1.3.2.2.1.1.3
Multiply by .
Step 1.3.2.2.1.2
Simplify by adding terms.
Step 1.3.2.2.1.2.1
Add and .
Step 1.3.2.2.1.2.2
Add and .
Step 1.3.3
Solve for in .
Step 1.3.3.1
Rewrite the equation as .
Step 1.3.3.2
Subtract from both sides of the equation.
Step 1.3.3.3
Divide each term in by and simplify.
Step 1.3.3.3.1
Divide each term in by .
Step 1.3.3.3.2
Simplify the left side.
Step 1.3.3.3.2.1
Cancel the common factor of .
Step 1.3.3.3.2.1.1
Cancel the common factor.
Step 1.3.3.3.2.1.2
Divide by .
Step 1.3.3.3.3
Simplify the right side.
Step 1.3.3.3.3.1
Cancel the common factor of and .
Step 1.3.3.3.3.1.1
Factor out of .
Step 1.3.3.3.3.1.2
Cancel the common factors.
Step 1.3.3.3.3.1.2.1
Factor out of .
Step 1.3.3.3.3.1.2.2
Cancel the common factor.
Step 1.3.3.3.3.1.2.3
Rewrite the expression.
Step 1.3.3.3.3.1.2.4
Divide by .
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
Simplify each term.
Step 1.3.4.2.1.1.1
Apply the distributive property.
Step 1.3.4.2.1.1.2
Cancel the common factor of .
Step 1.3.4.2.1.1.2.1
Factor out of .
Step 1.3.4.2.1.1.2.2
Factor out of .
Step 1.3.4.2.1.1.2.3
Cancel the common factor.
Step 1.3.4.2.1.1.2.4
Rewrite the expression.
Step 1.3.4.2.1.1.3
Multiply by .
Step 1.3.4.2.1.1.4
Rewrite as .
Step 1.3.4.2.1.2
Combine the opposite terms in .
Step 1.3.4.2.1.2.1
Subtract from .
Step 1.3.4.2.1.2.2
Add and .
Step 1.3.4.3
Replace all occurrences of in with .
Step 1.3.4.4
Simplify the right side.
Step 1.3.4.4.1
Simplify .
Step 1.3.4.4.1.1
Simplify each term.
Step 1.3.4.4.1.1.1
Apply the distributive property.
Step 1.3.4.4.1.1.2
Cancel the common factor of .
Step 1.3.4.4.1.1.2.1
Factor out of .
Step 1.3.4.4.1.1.2.2
Cancel the common factor.
Step 1.3.4.4.1.1.2.3
Rewrite the expression.
Step 1.3.4.4.1.1.3
Multiply by .
Step 1.3.4.4.1.2
Add and .
Step 1.3.4.5
Replace all occurrences of in with .
Step 1.3.4.6
Simplify the right side.
Step 1.3.4.6.1
Simplify .
Step 1.3.4.6.1.1
Simplify each term.
Step 1.3.4.6.1.1.1
Rewrite as .
Step 1.3.4.6.1.1.2
Apply the distributive property.
Step 1.3.4.6.1.1.3
Rewrite as .
Step 1.3.4.6.1.1.4
Multiply .
Step 1.3.4.6.1.1.4.1
Multiply by .
Step 1.3.4.6.1.1.4.2
Multiply by .
Step 1.3.4.6.1.2
Add and .
Step 1.3.5
Reorder and .
Step 1.3.6
Solve for in .
Step 1.3.6.1
Rewrite the equation as .
Step 1.3.6.2
Subtract from both sides of the equation.
Step 1.3.6.3
Divide each term in by and simplify.
Step 1.3.6.3.1
Divide each term in by .
Step 1.3.6.3.2
Simplify the left side.
Step 1.3.6.3.2.1
Dividing two negative values results in a positive value.
Step 1.3.6.3.2.2
Divide by .
Step 1.3.6.3.3
Simplify the right side.
Step 1.3.6.3.3.1
Simplify each term.
Step 1.3.6.3.3.1.1
Move the negative one from the denominator of .
Step 1.3.6.3.3.1.2
Rewrite as .
Step 1.3.6.3.3.1.3
Dividing two negative values results in a positive value.
Step 1.3.6.3.3.1.4
Divide by .
Step 1.3.7
Replace all occurrences of with in each equation.
Step 1.3.7.1
Replace all occurrences of in with .
Step 1.3.7.2
Simplify the right side.
Step 1.3.7.2.1
Simplify .
Step 1.3.7.2.1.1
Simplify each term.
Step 1.3.7.2.1.1.1
Apply the distributive property.
Step 1.3.7.2.1.1.2
Multiply by .
Step 1.3.7.2.1.1.3
Cancel the common factor of .
Step 1.3.7.2.1.1.3.1
Factor out of .
Step 1.3.7.2.1.1.3.2
Cancel the common factor.
Step 1.3.7.2.1.1.3.3
Rewrite the expression.
Step 1.3.7.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 1.3.7.2.1.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 1.3.7.2.1.3.1
Multiply by .
Step 1.3.7.2.1.3.2
Multiply by .
Step 1.3.7.2.1.4
Combine the numerators over the common denominator.
Step 1.3.7.2.1.5
Add and .
Step 1.3.7.3
Replace all occurrences of in with .
Step 1.3.7.4
Simplify the right side.
Step 1.3.7.4.1
Simplify .
Step 1.3.7.4.1.1
Simplify each term.
Step 1.3.7.4.1.1.1
Apply the distributive property.
Step 1.3.7.4.1.1.2
Multiply by .
Step 1.3.7.4.1.1.3
Cancel the common factor of .
Step 1.3.7.4.1.1.3.1
Factor out of .
Step 1.3.7.4.1.1.3.2
Cancel the common factor.
Step 1.3.7.4.1.1.3.3
Rewrite the expression.
Step 1.3.7.4.1.2
Add and .
Step 1.3.8
Solve for in .
Step 1.3.8.1
Rewrite the equation as .
Step 1.3.8.2
Move all terms not containing to the right side of the equation.
Step 1.3.8.2.1
Add to both sides of the equation.
Step 1.3.8.2.2
Subtract from both sides of the equation.
Step 1.3.8.3
Divide each term in by and simplify.
Step 1.3.8.3.1
Divide each term in by .
Step 1.3.8.3.2
Simplify the left side.
Step 1.3.8.3.2.1
Cancel the common factor of .
Step 1.3.8.3.2.1.1
Cancel the common factor.
Step 1.3.8.3.2.1.2
Divide by .
Step 1.3.8.3.3
Simplify the right side.
Step 1.3.8.3.3.1
Simplify each term.
Step 1.3.8.3.3.1.1
Cancel the common factor of and .
Step 1.3.8.3.3.1.1.1
Factor out of .
Step 1.3.8.3.3.1.1.2
Cancel the common factors.
Step 1.3.8.3.3.1.1.2.1
Factor out of .
Step 1.3.8.3.3.1.1.2.2
Cancel the common factor.
Step 1.3.8.3.3.1.1.2.3
Rewrite the expression.
Step 1.3.8.3.3.1.1.2.4
Divide by .
Step 1.3.8.3.3.1.2
Move the negative in front of the fraction.
Step 1.3.9
Replace all occurrences of with in each equation.
Step 1.3.9.1
Replace all occurrences of in with .
Step 1.3.9.2
Simplify the right side.
Step 1.3.9.2.1
Simplify .
Step 1.3.9.2.1.1
Simplify each term.
Step 1.3.9.2.1.1.1
Apply the distributive property.
Step 1.3.9.2.1.1.2
Multiply by .
Step 1.3.9.2.1.1.3
Multiply .
Step 1.3.9.2.1.1.3.1
Multiply by .
Step 1.3.9.2.1.1.3.2
Multiply by .
Step 1.3.9.2.1.2
Simplify terms.
Step 1.3.9.2.1.2.1
Combine the numerators over the common denominator.
Step 1.3.9.2.1.2.2
Add and .
Step 1.3.9.2.1.2.3
Cancel the common factor of and .
Step 1.3.9.2.1.2.3.1
Factor out of .
Step 1.3.9.2.1.2.3.2
Cancel the common factors.
Step 1.3.9.2.1.2.3.2.1
Factor out of .
Step 1.3.9.2.1.2.3.2.2
Cancel the common factor.
Step 1.3.9.2.1.2.3.2.3
Rewrite the expression.
Step 1.3.9.2.1.2.4
Subtract from .
Step 1.3.10
Solve for in .
Step 1.3.10.1
Rewrite the equation as .
Step 1.3.10.2
Subtract from both sides of the equation.
Step 1.3.10.3
Divide each term in by and simplify.
Step 1.3.10.3.1
Divide each term in by .
Step 1.3.10.3.2
Simplify the left side.
Step 1.3.10.3.2.1
Cancel the common factor of .
Step 1.3.10.3.2.1.1
Cancel the common factor.
Step 1.3.10.3.2.1.2
Divide by .
Step 1.3.10.3.3
Simplify the right side.
Step 1.3.10.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.3.10.3.3.2
Move the negative in front of the fraction.
Step 1.3.10.3.3.3
Multiply .
Step 1.3.10.3.3.3.1
Multiply by .
Step 1.3.10.3.3.3.2
Multiply by .
Step 1.3.10.3.3.3.3
Multiply by .
Step 1.3.10.3.3.3.4
Multiply by .
Step 1.3.11
Replace all occurrences of with in each equation.
Step 1.3.11.1
Replace all occurrences of in with .
Step 1.3.11.2
Simplify the right side.
Step 1.3.11.2.1
Simplify .
Step 1.3.11.2.1.1
Cancel the common factor of .
Step 1.3.11.2.1.1.1
Factor out of .
Step 1.3.11.2.1.1.2
Cancel the common factor.
Step 1.3.11.2.1.1.3
Rewrite the expression.
Step 1.3.11.2.1.2
Combine the numerators over the common denominator.
Step 1.3.11.2.1.3
Simplify the expression.
Step 1.3.11.2.1.3.1
Subtract from .
Step 1.3.11.2.1.3.2
Divide by .
Step 1.3.11.3
Replace all occurrences of in with .
Step 1.3.11.4
Simplify the right side.
Step 1.3.11.4.1
Simplify .
Step 1.3.11.4.1.1
To write as a fraction with a common denominator, multiply by .
Step 1.3.11.4.1.2
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 1.3.11.4.1.2.1
Multiply by .
Step 1.3.11.4.1.2.2
Multiply by .
Step 1.3.11.4.1.3
Combine the numerators over the common denominator.
Step 1.3.11.4.1.4
Add and .
Step 1.3.12
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
Simplify the numerator.
Step 1.5.1.1
Combine and .
Step 1.5.1.2
Add and .
Step 1.5.2
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.3
Multiply by .
Step 1.5.4
Move to the left of .
Step 1.5.5
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.6
Multiply by .
Step 1.5.7
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.8
Multiply by .
Step 2
Split the single integral into multiple integrals.
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Step 5.1
Let . Find .
Step 5.1.1
Differentiate .
Step 5.1.2
By the Sum Rule, the derivative of with respect to is .
Step 5.1.3
Evaluate .
Step 5.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3.2
Differentiate using the Power Rule which states that is where .
Step 5.1.3.3
Multiply by .
Step 5.1.4
Differentiate using the Constant Rule.
Step 5.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.4.2
Add and .
Step 5.2
Rewrite the problem using and .
Step 6
Step 6.1
Multiply by .
Step 6.2
Move to the left of .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Step 8.1
Multiply by .
Step 8.2
Multiply by .
Step 9
The integral of with respect to is .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Step 11.1
Let . Find .
Step 11.1.1
Differentiate .
Step 11.1.2
By the Sum Rule, the derivative of with respect to is .
Step 11.1.3
Evaluate .
Step 11.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 11.1.3.2
Differentiate using the Power Rule which states that is where .
Step 11.1.3.3
Multiply by .
Step 11.1.4
Differentiate using the Constant Rule.
Step 11.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 11.1.4.2
Add and .
Step 11.2
Rewrite the problem using and .
Step 12
Step 12.1
Multiply by .
Step 12.2
Move to the left of .
Step 13
Since is constant with respect to , move out of the integral.
Step 14
Step 14.1
Multiply by .
Step 14.2
Multiply by .
Step 15
The integral of with respect to is .
Step 16
Since is constant with respect to , move out of the integral.
Step 17
Step 17.1
Let . Find .
Step 17.1.1
Differentiate .
Step 17.1.2
By the Sum Rule, the derivative of with respect to is .
Step 17.1.3
Evaluate .
Step 17.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 17.1.3.2
Differentiate using the Power Rule which states that is where .
Step 17.1.3.3
Multiply by .
Step 17.1.4
Differentiate using the Constant Rule.
Step 17.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 17.1.4.2
Add and .
Step 17.2
Rewrite the problem using and .
Step 18
Step 18.1
Multiply by .
Step 18.2
Move to the left of .
Step 19
Since is constant with respect to , move out of the integral.
Step 20
Step 20.1
Multiply by .
Step 20.2
Multiply by .
Step 21
The integral of with respect to is .
Step 22
Simplify.
Step 23
Step 23.1
Replace all occurrences of with .
Step 23.2
Replace all occurrences of with .
Step 23.3
Replace all occurrences of with .