# Calculus Examples

Step 1

Step 1.1

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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Step 1.2

Divide the highest order term in the dividend by the highest order term in divisor .

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Step 1.3

Multiply the new quotient term by the divisor.

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Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in

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Step 1.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

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Step 1.6

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

Since is constant with respect to , move out of the integral.

Step 5

Step 5.1

Decompose the fraction and multiply through by the common denominator.

Step 5.1.1

Factor the fraction.

Step 5.1.1.1

Rewrite as .

Step 5.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, where and .

Step 5.1.2

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 5.1.3

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 5.1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .

Step 5.1.5

Cancel the common factor of .

Step 5.1.5.1

Cancel the common factor.

Step 5.1.5.2

Rewrite the expression.

Step 5.1.6

Cancel the common factor of .

Step 5.1.6.1

Cancel the common factor.

Step 5.1.6.2

Rewrite the expression.

Step 5.1.7

Simplify each term.

Step 5.1.7.1

Cancel the common factor of .

Step 5.1.7.1.1

Cancel the common factor.

Step 5.1.7.1.2

Divide by .

Step 5.1.7.2

Apply the distributive property.

Step 5.1.7.3

Move to the left of .

Step 5.1.7.4

Rewrite as .

Step 5.1.7.5

Cancel the common factor of .

Step 5.1.7.5.1

Cancel the common factor.

Step 5.1.7.5.2

Divide by .

Step 5.1.7.6

Apply the distributive property.

Step 5.1.7.7

Multiply by .

Step 5.1.8

Move .

Step 5.2

Create equations for the partial fraction variables and use them to set up a system of equations.

Step 5.2.1

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 5.2.2

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 5.2.3

Set up the system of equations to find the coefficients of the partial fractions.

Step 5.3

Solve the system of equations.

Step 5.3.1

Solve for in .

Step 5.3.1.1

Rewrite the equation as .

Step 5.3.1.2

Subtract from both sides of the equation.

Step 5.3.2

Replace all occurrences of with in each equation.

Step 5.3.2.1

Replace all occurrences of in with .

Step 5.3.2.2

Simplify the right side.

Step 5.3.2.2.1

Simplify .

Step 5.3.2.2.1.1

Multiply .

Step 5.3.2.2.1.1.1

Multiply by .

Step 5.3.2.2.1.1.2

Multiply by .

Step 5.3.2.2.1.2

Add and .

Step 5.3.3

Solve for in .

Step 5.3.3.1

Rewrite the equation as .

Step 5.3.3.2

Divide each term in by and simplify.

Step 5.3.3.2.1

Divide each term in by .

Step 5.3.3.2.2

Simplify the left side.

Step 5.3.3.2.2.1

Cancel the common factor of .

Step 5.3.3.2.2.1.1

Cancel the common factor.

Step 5.3.3.2.2.1.2

Divide by .

Step 5.3.4

Replace all occurrences of with in each equation.

Step 5.3.4.1

Replace all occurrences of in with .

Step 5.3.4.2

Simplify the right side.

Step 5.3.4.2.1

Multiply by .

Step 5.3.5

List all of the solutions.

Step 5.4

Replace each of the partial fraction coefficients in with the values found for and .

Step 5.5

Simplify.

Step 5.5.1

Multiply the numerator by the reciprocal of the denominator.

Step 5.5.2

Multiply by .

Step 5.5.3

Move to the left of .

Step 5.5.4

Multiply the numerator by the reciprocal of the denominator.

Step 5.5.5

Multiply by .

Step 6

Split the single integral into multiple integrals.

Step 7

Since is constant with respect to , move out of the integral.

Step 8

Since is constant with respect to , move out of the integral.

Step 9

Step 9.1

Let . Find .

Step 9.1.1

Differentiate .

Step 9.1.2

By the Sum Rule, the derivative of with respect to is .

Step 9.1.3

Differentiate using the Power Rule which states that is where .

Step 9.1.4

Since is constant with respect to , the derivative of with respect to is .

Step 9.1.5

Add and .

Step 9.2

Rewrite the problem using and .

Step 10

The integral of with respect to is .

Step 11

Since is constant with respect to , move out of the integral.

Step 12

Step 12.1

Let . Find .

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

Step 15.1

Replace all occurrences of with .

Step 15.2

Replace all occurrences of with .

Step 16

Step 16.1

Simplify each term.

Step 16.1.1

Combine and .

Step 16.1.2

Combine and .

Step 16.2

Combine the numerators over the common denominator.

Step 16.3

Cancel the common factor of .

Step 16.3.1

Cancel the common factor.

Step 16.3.2

Rewrite the expression.