Calculus Examples

Evaluate the Integral integral of (x^2)/(x^4-2x^2-8) with respect to x
Step 1
Write the fraction using partial fraction decomposition.
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Step 1.1
Decompose the fraction and multiply through by the common denominator.
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Step 1.1.1
Factor the fraction.
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Step 1.1.1.1
Rewrite as .
Step 1.1.1.2
Let . Substitute for all occurrences of .
Step 1.1.1.3
Factor using the AC method.
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Step 1.1.1.3.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 1.1.1.3.2
Write the factored form using these integers.
Step 1.1.1.4
Replace all occurrences of with .
Step 1.1.1.5
Rewrite as .
Step 1.1.1.6
Since both terms are perfect squares, factor using the difference of squares formula, where and .
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
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 1.1.5
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 .
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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 .
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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 .
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Step 1.1.8.1
Cancel the common factor.
Step 1.1.8.2
Divide by .
Step 1.1.9
Simplify each term.
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Step 1.1.9.1
Cancel the common factor of .
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Step 1.1.9.1.1
Cancel the common factor.
Step 1.1.9.1.2
Divide by .
Step 1.1.9.2
Apply the distributive property.
Step 1.1.9.3
Move to the left of .
Step 1.1.9.4
Expand using the FOIL Method.
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Step 1.1.9.4.1
Apply the distributive property.
Step 1.1.9.4.2
Apply the distributive property.
Step 1.1.9.4.3
Apply the distributive property.
Step 1.1.9.5
Simplify each term.
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Step 1.1.9.5.1
Multiply by by adding the exponents.
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Step 1.1.9.5.1.1
Move .
Step 1.1.9.5.1.2
Multiply by .
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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
Move to the left of .
Step 1.1.9.5.3
Multiply by .
Step 1.1.9.6
Cancel the common factor of .
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Step 1.1.9.6.1
Cancel the common factor.
Step 1.1.9.6.2
Divide by .
Step 1.1.9.7
Apply the distributive property.
Step 1.1.9.8
Move to the left of .
Step 1.1.9.9
Expand using the FOIL Method.
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Step 1.1.9.9.1
Apply the distributive property.
Step 1.1.9.9.2
Apply the distributive property.
Step 1.1.9.9.3
Apply the distributive property.
Step 1.1.9.10
Simplify each term.
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Step 1.1.9.10.1
Multiply by by adding the exponents.
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Step 1.1.9.10.1.1
Move .
Step 1.1.9.10.1.2
Multiply by .
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Step 1.1.9.10.1.2.1
Raise to the power of .
Step 1.1.9.10.1.2.2
Use the power rule to combine exponents.
Step 1.1.9.10.1.3
Add and .
Step 1.1.9.10.2
Move to the left of .
Step 1.1.9.10.3
Multiply by .
Step 1.1.9.11
Cancel the common factor of .
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Step 1.1.9.11.1
Cancel the common factor.
Step 1.1.9.11.2
Divide by .
Step 1.1.9.12
Expand using the FOIL Method.
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Step 1.1.9.12.1
Apply the distributive property.
Step 1.1.9.12.2
Apply the distributive property.
Step 1.1.9.12.3
Apply the distributive property.
Step 1.1.9.13
Simplify each term.
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Step 1.1.9.13.1
Multiply by by adding the exponents.
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Step 1.1.9.13.1.1
Move .
Step 1.1.9.13.1.2
Multiply by .
Step 1.1.9.13.2
Move to the left of .
Step 1.1.9.13.3
Move to the left of .
Step 1.1.9.14
Expand by multiplying each term in the first expression by each term in the second expression.
Step 1.1.9.15
Combine the opposite terms in .
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Step 1.1.9.15.1
Reorder the factors in the terms and .
Step 1.1.9.15.2
Add and .
Step 1.1.9.15.3
Add and .
Step 1.1.9.16
Simplify each term.
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Step 1.1.9.16.1
Multiply by by adding the exponents.
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Step 1.1.9.16.1.1
Move .
Step 1.1.9.16.1.2
Multiply by .
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Step 1.1.9.16.1.2.1
Raise to the power of .
Step 1.1.9.16.1.2.2
Use the power rule to combine exponents.
Step 1.1.9.16.1.3
Add and .
Step 1.1.9.16.2
Move to the left of .
Step 1.1.9.16.3
Multiply by by adding the exponents.
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Step 1.1.9.16.3.1
Move .
Step 1.1.9.16.3.2
Multiply by .
Step 1.1.9.16.4
Multiply by .
Step 1.1.9.16.5
Multiply by by adding the exponents.
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Step 1.1.9.16.5.1
Move .
Step 1.1.9.16.5.2
Multiply by .
Step 1.1.9.16.6
Multiply by .
Step 1.1.9.17
Combine the opposite terms in .
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Step 1.1.9.17.1
Add and .
Step 1.1.9.17.2
Add and .
Step 1.1.10
Simplify the expression.
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Step 1.1.10.1
Move .
Step 1.1.10.2
Move .
Step 1.1.10.3
Reorder and .
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 1.1.10.10
Move .
Step 1.1.10.11
Move .
Step 1.1.10.12
Move .
Step 1.2
Create equations for the partial fraction variables and use them to set up a system of equations.
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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.
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Step 1.3.1
Solve for in .
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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.
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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 1.3.2
Replace all occurrences of with in each equation.
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Step 1.3.2.1
Replace all occurrences of in with .
Step 1.3.2.2
Simplify the right side.
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Step 1.3.2.2.1
Simplify .
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Step 1.3.2.2.1.1
Simplify each term.
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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
Add and .
Step 1.3.2.3
Replace all occurrences of in with .
Step 1.3.2.4
Simplify the right side.
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Step 1.3.2.4.1
Simplify .
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Step 1.3.2.4.1.1
Simplify each term.
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Step 1.3.2.4.1.1.1
Apply the distributive property.
Step 1.3.2.4.1.1.2
Multiply by .
Step 1.3.2.4.1.1.3
Multiply by .
Step 1.3.2.4.1.2
Simplify by adding terms.
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Step 1.3.2.4.1.2.1
Combine the opposite terms in .
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Step 1.3.2.4.1.2.1.1
Add and .
Step 1.3.2.4.1.2.1.2
Add and .
Step 1.3.2.4.1.2.2
Subtract from .
Step 1.3.2.5
Replace all occurrences of in with .
Step 1.3.2.6
Simplify the right side.
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Step 1.3.2.6.1
Simplify .
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Step 1.3.2.6.1.1
Simplify each term.
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Step 1.3.2.6.1.1.1
Apply the distributive property.
Step 1.3.2.6.1.1.2
Multiply by .
Step 1.3.2.6.1.1.3
Multiply by .
Step 1.3.2.6.1.2
Add and .
Step 1.3.3
Solve for in .
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Step 1.3.3.1
Rewrite the equation as .
Step 1.3.3.2
Divide each term in by and simplify.
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Step 1.3.3.2.1
Divide each term in by .
Step 1.3.3.2.2
Simplify the left side.
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Step 1.3.3.2.2.1
Cancel the common factor of .
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Step 1.3.3.2.2.1.1
Cancel the common factor.
Step 1.3.3.2.2.1.2
Divide by .
Step 1.3.3.2.3
Simplify the right side.
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Step 1.3.3.2.3.1
Divide by .
Step 1.3.4
Replace all occurrences of with in each equation.
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Step 1.3.4.1
Replace all occurrences of in with .
Step 1.3.4.2
Simplify the right side.
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Step 1.3.4.2.1
Simplify .
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Step 1.3.4.2.1.1
Multiply by .
Step 1.3.4.2.1.2
Add and .
Step 1.3.4.3
Replace all occurrences of in with .
Step 1.3.4.4
Simplify the right side.
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Step 1.3.4.4.1
Simplify .
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Step 1.3.4.4.1.1
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.
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Step 1.3.4.6.1
Subtract from .
Step 1.3.5
Solve for in .
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Step 1.3.5.1
Rewrite the equation as .
Step 1.3.5.2
Subtract from both sides of the equation.
Step 1.3.6
Replace all occurrences of with in each equation.
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Step 1.3.6.1
Replace all occurrences of in with .
Step 1.3.6.2
Simplify the right side.
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Step 1.3.6.2.1
Simplify .
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Step 1.3.6.2.1.1
Simplify each term.
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Step 1.3.6.2.1.1.1
Apply the distributive property.
Step 1.3.6.2.1.1.2
Multiply by .
Step 1.3.6.2.1.1.3
Multiply by .
Step 1.3.6.2.1.2
Add and .
Step 1.3.7
Solve for in .
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Step 1.3.7.1
Rewrite the equation as .
Step 1.3.7.2
Add to both sides of the equation.
Step 1.3.7.3
Divide each term in by and simplify.
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Step 1.3.7.3.1
Divide each term in by .
Step 1.3.7.3.2
Simplify the left side.
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Step 1.3.7.3.2.1
Cancel the common factor of .
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Step 1.3.7.3.2.1.1
Cancel the common factor.
Step 1.3.7.3.2.1.2
Divide by .
Step 1.3.7.3.3
Simplify the right side.
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Step 1.3.7.3.3.1
Cancel the common factor of and .
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Step 1.3.7.3.3.1.1
Factor out of .
Step 1.3.7.3.3.1.2
Cancel the common factors.
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Step 1.3.7.3.3.1.2.1
Factor out of .
Step 1.3.7.3.3.1.2.2
Cancel the common factor.
Step 1.3.7.3.3.1.2.3
Rewrite the expression.
Step 1.3.8
Replace all occurrences of with in each equation.
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Step 1.3.8.1
Replace all occurrences of in with .
Step 1.3.8.2
Simplify the right side.
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Step 1.3.8.2.1
Simplify .
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Step 1.3.8.2.1.1
Simplify each term.
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Step 1.3.8.2.1.1.1
Cancel the common factor of .
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Step 1.3.8.2.1.1.1.1
Factor out of .
Step 1.3.8.2.1.1.1.2
Factor out of .
Step 1.3.8.2.1.1.1.3
Cancel the common factor.
Step 1.3.8.2.1.1.1.4
Rewrite the expression.
Step 1.3.8.2.1.1.2
Combine and .
Step 1.3.8.2.1.1.3
Move the negative in front of the fraction.
Step 1.3.8.2.1.2
Simplify the expression.
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Step 1.3.8.2.1.2.1
Write as a fraction with a common denominator.
Step 1.3.8.2.1.2.2
Combine the numerators over the common denominator.
Step 1.3.8.2.1.2.3
Subtract from .
Step 1.3.8.3
Replace all occurrences of in with .
Step 1.3.8.4
Simplify the right side.
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Step 1.3.8.4.1
Multiply by .
Step 1.3.9
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.
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Step 1.5.1
Multiply the numerator and denominator of the fraction by .
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Step 1.5.1.1
Multiply by .
Step 1.5.1.2
Combine.
Step 1.5.2
Apply the distributive property.
Step 1.5.3
Cancel the common factor of .
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Step 1.5.3.1
Cancel the common factor.
Step 1.5.3.2
Rewrite the expression.
Step 1.5.4
Simplify the numerator.
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Step 1.5.4.1
Multiply by .
Step 1.5.4.2
Multiply by .
Step 1.5.4.3
Add and .
Step 1.5.5
Factor out of .
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Step 1.5.5.1
Factor out of .
Step 1.5.5.2
Factor out of .
Step 1.5.5.3
Factor out of .
Step 1.5.6
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.7
Multiply by .
Step 1.5.8
Move to the left of .
Step 1.5.9
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.10
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
Let . Then . Rewrite using and .
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Step 5.1
Let . Find .
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Step 5.1.1
Differentiate .
Step 5.1.2
By the Sum Rule, the derivative of with respect to is .
Step 5.1.3
Differentiate using the Power Rule which states that is where .
Step 5.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.5
Add and .
Step 5.2
Rewrite the problem using and .
Step 6
The integral of with respect to is .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Let . Then . Rewrite using and .
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Step 8.1
Let . Find .
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Step 8.1.1
Differentiate .
Step 8.1.2
By the Sum Rule, the derivative of with respect to is .
Step 8.1.3
Differentiate using the Power Rule which states that is where .
Step 8.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 8.1.5
Add and .
Step 8.2
Rewrite the problem using and .
Step 9
The integral of with respect to is .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Reorder and .
Step 12
Rewrite as .
Step 13
The integral of with respect to is .
Step 14
Simplify.
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Step 14.1
Combine and .
Step 14.2
Simplify.
Step 14.3
Multiply by .
Step 15
Substitute back in for each integration substitution variable.
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Step 15.1
Replace all occurrences of with .
Step 15.2
Replace all occurrences of with .