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# Calculus Examples

Step 1

Step 1.1

Let . Find .

Step 1.1.1

Differentiate .

Step 1.1.2

Differentiate.

Step 1.1.2.1

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

Step 1.1.2.2

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

Step 1.1.3

Evaluate .

Step 1.1.3.1

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

Step 1.1.3.2

Differentiate using the Power Rule which states that is where .

Step 1.1.3.3

Multiply by .

Step 1.1.4

Add and .

Step 1.2

Substitute the lower limit in for in .

Step 1.3

Simplify.

Step 1.3.1

Multiply by .

Step 1.3.2

Add and .

Step 1.4

Substitute the upper limit in for in .

Step 1.5

Simplify.

Step 1.5.1

Multiply by .

Step 1.5.2

Add and .

Step 1.6

The values found for and will be used to evaluate the definite integral.

Step 1.7

Rewrite the problem using , , and the new limits of integration.

Step 2

Combine and .

Step 3

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

Step 4

Use to rewrite as .

Step 5

By the Power Rule, the integral of with respect to is .

Step 6

Step 6.1

Evaluate at and at .

Step 6.2

Simplify.

Step 6.2.1

Rewrite as .

Step 6.2.2

Apply the power rule and multiply exponents, .

Step 6.2.3

Cancel the common factor of .

Step 6.2.3.1

Cancel the common factor.

Step 6.2.3.2

Rewrite the expression.

Step 6.2.4

Raise to the power of .

Step 6.2.5

Combine and .

Step 6.2.6

Multiply by .

Step 6.2.7

Cancel the common factor of and .

Step 6.2.7.1

Factor out of .

Step 6.2.7.2

Cancel the common factors.

Step 6.2.7.2.1

Factor out of .

Step 6.2.7.2.2

Cancel the common factor.

Step 6.2.7.2.3

Rewrite the expression.

Step 6.2.7.2.4

Divide by .

Step 6.2.8

One to any power is one.

Step 6.2.9

Multiply by .

Step 6.2.10

To write as a fraction with a common denominator, multiply by .

Step 6.2.11

Combine and .

Step 6.2.12

Combine the numerators over the common denominator.

Step 6.2.13

Simplify the numerator.

Step 6.2.13.1

Multiply by .

Step 6.2.13.2

Subtract from .

Step 6.2.14

Multiply by .

Step 6.2.15

Multiply by .

Step 7

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Mixed Number Form:

Step 8