Calculus Examples

Step 1
Divide by .
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Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
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Divide the highest order term in the dividend by the highest order term in divisor .
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Multiply the new quotient term by the divisor.
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The expression needs to be subtracted from the dividend, so change all the signs in
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After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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The final answer is the quotient plus the remainder over the divisor.
Step 2
Split the single integral into multiple integrals.
Step 3
Apply the constant rule.
Step 4
Write the fraction using partial fraction decomposition.
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Decompose the fraction and multiply through by the common denominator.
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Factor the fraction.
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Rewrite as .
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Simplify.
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Multiply by .
One to any power is one.
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 .
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.
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify each term.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Apply the distributive property.
Multiply by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify each term.
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Multiply by by adding the exponents.
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Move .
Multiply by .
Move to the left of .
Rewrite as .
Move to the left of .
Rewrite as .
Simplify the expression.
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Reorder and .
Move .
Reorder and .
Move .
Move .
Create equations for the partial fraction variables and use them to set up a system of equations.
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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.
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.
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.
Set up the system of equations to find the coefficients of the partial fractions.
Solve the system of equations.
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Solve for in .
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Rewrite the equation as .
Subtract from both sides of the equation.
Replace all occurrences of with in each equation.
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Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Rewrite as .
Subtract from .
Replace all occurrences of in with .
Simplify the right side.
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Rewrite as .
Solve for in .
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Rewrite the equation as .
Add to both sides of the equation.
Replace all occurrences of with in each equation.
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Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Solve for in .
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Rewrite the equation as .
Move all terms not containing to the right side of the equation.
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Add to both sides of the equation.
Add and .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Move the negative in front of the fraction.
Replace all occurrences of with in each equation.
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Replace all occurrences of in with .
Simplify the right side.
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Simplify .
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Simplify each term.
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Multiply .
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Multiply by .
Combine and .
Multiply by .
Move the negative in front of the fraction.
Simplify the expression.
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Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Subtract from .
Move the negative in front of the fraction.
Replace all occurrences of in with .
Simplify the right side.
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Multiply .
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Multiply by .
Multiply by .
List all of the solutions.
Replace each of the partial fraction coefficients in with the values found for , , and .
Simplify.
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Multiply the numerator and denominator of the complex fraction by .
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Multiply by .
Combine.
Apply the distributive property.
Cancel the common factor of .
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Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Simplify the numerator.
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Combine and .
Cancel the common factor of .
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Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Multiply by .
Simplify with factoring out.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Rewrite as .
Factor out of .
Simplify the expression.
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Rewrite as .
Move the negative in front of the fraction.
Multiply the numerator by the reciprocal of the denominator.
Multiply by .
Step 5
Split the single integral into multiple integrals.
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Let . Then . Rewrite using and .
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Let . Find .
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Differentiate .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Add and .
Rewrite the problem using 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
Let . Then , so . Rewrite using and .
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Let . Find .
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Differentiate .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Add and .
Rewrite the problem using and .
Step 12
The integral of with respect to is .
Step 13
Simplify.
Step 14
Substitute back in for each integration substitution variable.
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Replace all occurrences of with .
Replace all occurrences of with .
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