# Calculus Examples

Step 1

Step 1.1

Decompose the fraction and multiply through by the common denominator.

Step 1.1.1

Factor using the AC method.

Step 1.1.1.1

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Step 1.1.1.2

Write the factored form using these integers.

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 .

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 .

Step 1.1.6.1

Cancel the common factor.

Step 1.1.6.2

Divide by .

Step 1.1.7

Simplify each term.

Step 1.1.7.1

Cancel the common factor of .

Step 1.1.7.1.1

Cancel the common factor.

Step 1.1.7.1.2

Divide by .

Step 1.1.7.2

Apply the distributive property.

Step 1.1.7.3

Move to the left of .

Step 1.1.7.4

Cancel the common factor of .

Step 1.1.7.4.1

Cancel the common factor.

Step 1.1.7.4.2

Divide by .

Step 1.1.7.5

Apply the distributive property.

Step 1.1.7.6

Move to the left of .

Step 1.1.7.7

Rewrite as .

Step 1.1.8

Move .

Step 1.2

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

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.

Step 1.3.1

Solve for in .

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.

Step 1.3.2.1

Replace all occurrences of in with .

Step 1.3.2.2

Simplify the right side.

Step 1.3.2.2.1

Simplify .

Step 1.3.2.2.1.1

Simplify each term.

Step 1.3.2.2.1.1.1

Apply the distributive property.

Step 1.3.2.2.1.1.2

Multiply by .

Step 1.3.2.2.1.1.3

Multiply by .

Step 1.3.2.2.1.1.4

Rewrite as .

Step 1.3.2.2.1.2

Subtract from .

Step 1.3.3

Solve for in .

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.

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.

Step 1.3.3.3.1

Divide each term in by .

Step 1.3.3.3.2

Simplify the left side.

Step 1.3.3.3.2.1

Cancel the common factor of .

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.

Step 1.3.3.3.3.1

Divide by .

Step 1.3.4

Replace all occurrences of with in each equation.

Step 1.3.4.1

Replace all occurrences of in with .

Step 1.3.4.2

Simplify the right side.

Step 1.3.4.2.1

Simplify .

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

Step 4.1

Let . Find .

Step 4.1.1

Differentiate .

Step 4.1.2

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

Step 4.1.3

Differentiate using the Power Rule which states that is where .

Step 4.1.4

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

Step 4.1.5

Add and .

Step 4.2

Rewrite the problem using and .

Step 5

The integral of with respect to is .

Step 6

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

Step 7

Step 7.1

Let . Find .

Step 7.1.1

Differentiate .

Step 7.1.2

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

Step 7.1.3

Differentiate using the Power Rule which states that is where .

Step 7.1.4

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

Step 7.1.5

Add and .

Step 7.2

Rewrite the problem using and .

Step 8

The integral of with respect to is .

Step 9

Simplify.

Step 10

Step 10.1

Replace all occurrences of with .

Step 10.2

Replace all occurrences of with .