Calculus Examples

Find the Integral (-x^3+17x^2-12x-9)/(x^4-3x^3)
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 out of .
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Step 1.1.1.1
Factor out of .
Step 1.1.1.2
Factor out of .
Step 1.1.1.3
Factor out of .
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
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 1.1.4
Cancel the common factor of .
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Step 1.1.4.1
Cancel the common factor.
Step 1.1.4.2
Rewrite the expression.
Step 1.1.5
Cancel the common factor of .
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Step 1.1.5.1
Cancel the common factor.
Step 1.1.5.2
Divide by .
Step 1.1.6
Simplify each term.
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Step 1.1.6.1
Cancel the common factor of .
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Step 1.1.6.1.1
Cancel the common factor.
Step 1.1.6.1.2
Divide by .
Step 1.1.6.2
Apply the distributive property.
Step 1.1.6.3
Move to the left of .
Step 1.1.6.4
Cancel the common factor of and .
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Step 1.1.6.4.1
Factor out of .
Step 1.1.6.4.2
Cancel the common factors.
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Step 1.1.6.4.2.1
Multiply by .
Step 1.1.6.4.2.2
Cancel the common factor.
Step 1.1.6.4.2.3
Rewrite the expression.
Step 1.1.6.4.2.4
Divide by .
Step 1.1.6.5
Apply the distributive property.
Step 1.1.6.6
Multiply by .
Step 1.1.6.7
Move to the left of .
Step 1.1.6.8
Apply the distributive property.
Step 1.1.6.9
Rewrite using the commutative property of multiplication.
Step 1.1.6.10
Cancel the common factor of and .
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Step 1.1.6.10.1
Factor out of .
Step 1.1.6.10.2
Cancel the common factors.
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Step 1.1.6.10.2.1
Raise to the power of .
Step 1.1.6.10.2.2
Factor out of .
Step 1.1.6.10.2.3
Cancel the common factor.
Step 1.1.6.10.2.4
Rewrite the expression.
Step 1.1.6.10.2.5
Divide by .
Step 1.1.6.11
Apply the distributive property.
Step 1.1.6.12
Multiply by by adding the exponents.
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Step 1.1.6.12.1
Multiply by .
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Step 1.1.6.12.1.1
Raise to the power of .
Step 1.1.6.12.1.2
Use the power rule to combine exponents.
Step 1.1.6.12.2
Add and .
Step 1.1.6.13
Move to the left of .
Step 1.1.6.14
Apply the distributive property.
Step 1.1.6.15
Rewrite using the commutative property of multiplication.
Step 1.1.6.16
Cancel the common factor of .
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Step 1.1.6.16.1
Cancel the common factor.
Step 1.1.6.16.2
Divide by .
Step 1.1.7
Simplify the expression.
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Step 1.1.7.1
Move .
Step 1.1.7.2
Move .
Step 1.1.7.3
Move .
Step 1.1.7.4
Move .
Step 1.1.7.5
Move .
Step 1.1.7.6
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
Divide each term in by and simplify.
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Step 1.3.1.2.1
Divide each term in by .
Step 1.3.1.2.2
Simplify the left side.
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Step 1.3.1.2.2.1
Cancel the common factor of .
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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.
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Step 1.3.1.2.3.1
Divide by .
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
Remove parentheses.
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
Move all terms not containing to the right side of the equation.
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Step 1.3.3.2.1
Subtract from both sides of the equation.
Step 1.3.3.2.2
Subtract from .
Step 1.3.3.3
Divide each term in by and simplify.
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Step 1.3.3.3.1
Divide each term in by .
Step 1.3.3.3.2
Simplify the left side.
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Step 1.3.3.3.2.1
Cancel the common factor of .
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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.
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Step 1.3.3.3.3.1
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
Remove parentheses.
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
Move all terms not containing to the right side of the equation.
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Step 1.3.5.2.1
Subtract from both sides of the equation.
Step 1.3.5.2.2
Subtract from .
Step 1.3.5.3
Divide each term in by and simplify.
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Step 1.3.5.3.1
Divide each term in by .
Step 1.3.5.3.2
Simplify the left side.
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Step 1.3.5.3.2.1
Cancel the common factor of .
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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.
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Step 1.3.5.3.3.1
Divide by .
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
Remove parentheses.
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
Move all terms not containing to the right side of the equation.
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Step 1.3.7.2.1
Add to both sides of the equation.
Step 1.3.7.2.2
Add and .
Step 1.3.8
Solve the system of equations.
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
Move the negative in front of the fraction.
Step 2
Split the single integral into multiple integrals.
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Apply basic rules of exponents.
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Step 4.1
Move out of the denominator by raising it to the power.
Step 4.2
Multiply the exponents in .
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Step 4.2.1
Apply the power rule and multiply exponents, .
Step 4.2.2
Multiply by .
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Simplify.
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Step 6.1
Combine and .
Step 6.2
Move to the denominator using the negative exponent rule .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Apply basic rules of exponents.
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Step 8.1
Move out of the denominator by raising it to the power.
Step 8.2
Multiply the exponents in .
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Step 8.2.1
Apply the power rule and multiply exponents, .
Step 8.2.2
Multiply by .
Step 9
By the Power Rule, the integral of with respect to is .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Since is constant with respect to , move out of the integral.
Step 12
Multiply by .
Step 13
The integral of with respect to is .
Step 14
Since is constant with respect to , move out of the integral.
Step 15
Let . Then . Rewrite using and .
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Step 15.1
Let . Find .
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Step 15.1.1
Differentiate .
Step 15.1.2
By the Sum Rule, the derivative of with respect to is .
Step 15.1.3
Differentiate using the Power Rule which states that is where .
Step 15.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 15.1.5
Add and .
Step 15.2
Rewrite the problem using and .
Step 16
The integral of with respect to is .
Step 17
Simplify.
Step 18
Replace all occurrences of with .