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Calculus Examples
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Step 2.1
Decompose the fraction and multiply through by the common denominator.
Step 2.1.1
Factor by grouping.
Step 2.1.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 2.1.1.1.1
Factor out of .
Step 2.1.1.1.2
Rewrite as plus
Step 2.1.1.1.3
Apply the distributive property.
Step 2.1.1.1.4
Multiply by .
Step 2.1.1.2
Factor out the greatest common factor from each group.
Step 2.1.1.2.1
Group the first two terms and the last two terms.
Step 2.1.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 2.1.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 2.1.2
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 2.1.3
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 2.1.4
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 2.1.5
Cancel the common factor of .
Step 2.1.5.1
Cancel the common factor.
Step 2.1.5.2
Rewrite the expression.
Step 2.1.6
Cancel the common factor of .
Step 2.1.6.1
Cancel the common factor.
Step 2.1.6.2
Rewrite the expression.
Step 2.1.7
Simplify each term.
Step 2.1.7.1
Cancel the common factor of .
Step 2.1.7.1.1
Cancel the common factor.
Step 2.1.7.1.2
Divide by .
Step 2.1.7.2
Apply the distributive property.
Step 2.1.7.3
Multiply by .
Step 2.1.7.4
Cancel the common factor of .
Step 2.1.7.4.1
Cancel the common factor.
Step 2.1.7.4.2
Divide by .
Step 2.1.7.5
Apply the distributive property.
Step 2.1.7.6
Rewrite using the commutative property of multiplication.
Step 2.1.7.7
Multiply by .
Step 2.1.8
Simplify the expression.
Step 2.1.8.1
Move .
Step 2.1.8.2
Move .
Step 2.2
Create equations for the partial fraction variables and use them to set up a system of equations.
Step 2.2.1
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 2.2.2
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 2.2.3
Set up the system of equations to find the coefficients of the partial fractions.
Step 2.3
Solve the system of equations.
Step 2.3.1
Solve for in .
Step 2.3.1.1
Rewrite the equation as .
Step 2.3.1.2
Subtract from both sides of the equation.
Step 2.3.2
Replace all occurrences of with in each equation.
Step 2.3.2.1
Replace all occurrences of in with .
Step 2.3.2.2
Simplify the right side.
Step 2.3.2.2.1
Add and .
Step 2.3.3
Solve for in .
Step 2.3.3.1
Rewrite the equation as .
Step 2.3.3.2
Divide each term in by and simplify.
Step 2.3.3.2.1
Divide each term in by .
Step 2.3.3.2.2
Simplify the left side.
Step 2.3.3.2.2.1
Dividing two negative values results in a positive value.
Step 2.3.3.2.2.2
Divide by .
Step 2.3.3.2.3
Simplify the right side.
Step 2.3.3.2.3.1
Divide by .
Step 2.3.4
Replace all occurrences of with in each equation.
Step 2.3.4.1
Replace all occurrences of in with .
Step 2.3.4.2
Simplify the right side.
Step 2.3.4.2.1
Multiply by .
Step 2.3.5
List all of the solutions.
Step 2.4
Replace each of the partial fraction coefficients in with the values found for and .
Step 2.5
Move the negative in front of the fraction.
Step 3
Split the single integral into multiple integrals.
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
Substitute the lower limit in for in .
Step 5.3
Simplify.
Step 5.3.1
Multiply by .
Step 5.3.2
Add and .
Step 5.4
Substitute the upper limit in for in .
Step 5.5
Simplify.
Step 5.5.1
Multiply by .
Step 5.5.2
Add and .
Step 5.6
The values found for and will be used to evaluate the definite integral.
Step 5.7
Rewrite the problem using , , and the new limits of integration.
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
Combine and .
Step 8.2
Cancel the common factor of .
Step 8.2.1
Cancel the common factor.
Step 8.2.2
Rewrite the expression.
Step 8.3
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
Differentiate using the Power Rule which states that is where .
Step 11.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 11.1.5
Add and .
Step 11.2
Substitute the lower limit in for in .
Step 11.3
Add and .
Step 11.4
Substitute the upper limit in for in .
Step 11.5
Add and .
Step 11.6
The values found for and will be used to evaluate the definite integral.
Step 11.7
Rewrite the problem using , , and the new limits of integration.
Step 12
The integral of with respect to is .
Step 13
Step 13.1
Evaluate at and at .
Step 13.2
Evaluate at and at .
Step 13.3
Remove parentheses.
Step 14
Step 14.1
Use the quotient property of logarithms, .
Step 14.2
Use the quotient property of logarithms, .
Step 14.3
Use the quotient property of logarithms, .
Step 14.4
Rewrite as a product.
Step 14.5
Multiply by the reciprocal of the fraction to divide by .
Step 14.6
Multiply by .
Step 14.7
Multiply by .
Step 14.8
To multiply absolute values, multiply the terms inside each absolute value.
Step 14.9
Multiply by .
Step 14.10
To multiply absolute values, multiply the terms inside each absolute value.
Step 14.11
Multiply by .
Step 15
Step 15.1
The absolute value is the distance between a number and zero. The distance between and is .
Step 15.2
The absolute value is the distance between a number and zero. The distance between and is .
Step 16
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 17