Calculus Examples

Evaluate the Integral integral of (5-x)/(2x^2+x-1) 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 by grouping.
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Step 1.1.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 1.1.1.1.1
Multiply by .
Step 1.1.1.1.2
Rewrite as plus
Step 1.1.1.1.3
Apply the distributive property.
Step 1.1.1.2
Factor out the greatest common factor from each group.
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Step 1.1.1.2.1
Group the first two terms and the last two terms.
Step 1.1.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 1.1.1.3
Factor the polynomial by factoring out the greatest common factor, .
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 .
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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 .
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Step 1.1.6.1
Cancel the common factor.
Step 1.1.6.2
Divide by .
Step 1.1.7
Reorder and .
Step 1.1.8
Simplify each term.
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Step 1.1.8.1
Cancel the common factor of .
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Step 1.1.8.1.1
Cancel the common factor.
Step 1.1.8.1.2
Divide by .
Step 1.1.8.2
Apply the distributive property.
Step 1.1.8.3
Multiply by .
Step 1.1.8.4
Cancel the common factor of .
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Step 1.1.8.4.1
Cancel the common factor.
Step 1.1.8.4.2
Divide by .
Step 1.1.8.5
Apply the distributive property.
Step 1.1.8.6
Rewrite using the commutative property of multiplication.
Step 1.1.8.7
Move to the left of .
Step 1.1.8.8
Rewrite as .
Step 1.1.9
Simplify the expression.
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Step 1.1.9.1
Move .
Step 1.1.9.2
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 the terms not containing . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 1.2.3
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
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
Rewrite as .
Step 1.3.2.2.1.2
Subtract from .
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
Add to both sides of the equation.
Step 1.3.3.2.2
Add and .
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
Simplify .
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Step 1.3.4.2.1.1
Multiply by .
Step 1.3.4.2.1.2
Add and .
Step 1.3.5
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
Let . Then , so . Rewrite using and .
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Step 4.1
Let . Find .
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Step 4.1.1
Differentiate .
Step 4.1.2
By the Sum Rule, the derivative of with respect to is .
Step 4.1.3
Evaluate .
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Step 4.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.2
Differentiate using the Power Rule which states that is where .
Step 4.1.3.3
Multiply by .
Step 4.1.4
Differentiate using the Constant Rule.
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Step 4.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.4.2
Add and .
Step 4.2
Rewrite the problem using and .
Step 5
Simplify.
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Step 5.1
Multiply by .
Step 5.2
Move to the left of .
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Combine and .
Step 8
The integral of with respect to is .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Multiply by .
Step 12
Let . Then . Rewrite using and .
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Step 12.1
Let . Find .
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Step 12.1.1
Differentiate .
Step 12.1.2
By the Sum Rule, the derivative of with respect to is .
Step 12.1.3
Differentiate using the Power Rule which states that is where .
Step 12.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 12.1.5
Add and .
Step 12.2
Rewrite the problem using and .
Step 13
The integral of with respect to is .
Step 14
Simplify.
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 .