# Calculus Examples

Step 1

Let , where . Then . Note that since , is positive.

Step 2

Step 2.1

Simplify .

Step 2.1.1

Simplify each term.

Step 2.1.1.1

Apply the product rule to .

Step 2.1.1.2

Raise to the power of .

Step 2.1.1.3

Multiply by .

Step 2.1.2

Factor out of .

Step 2.1.3

Factor out of .

Step 2.1.4

Factor out of .

Step 2.1.5

Apply pythagorean identity.

Step 2.1.6

Rewrite as .

Step 2.1.7

Pull terms out from under the radical, assuming positive real numbers.

Step 2.2

Simplify.

Step 2.2.1

Multiply by .

Step 2.2.2

Raise to the power of .

Step 2.2.3

Raise to the power of .

Step 2.2.4

Use the power rule to combine exponents.

Step 2.2.5

Add and .

Step 3

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

Step 4

Use the half-angle formula to rewrite as .

Step 5

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

Step 6

Combine and .

Step 7

Split the single integral into multiple integrals.

Step 8

Apply the constant rule.

Step 9

Step 9.1

Let . Find .

Step 9.1.1

Differentiate .

Step 9.1.2

Since is constant with respect to , 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

Multiply by .

Step 9.2

Rewrite the problem using and .

Step 10

Combine and .

Step 11

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

Step 12

The integral of with respect to is .

Step 13

Simplify.

Step 14

Step 14.1

Replace all occurrences of with .

Step 14.2

Replace all occurrences of with .

Step 14.3

Replace all occurrences of with .

Step 15

Step 15.1

Combine and .

Step 15.2

Apply the distributive property.

Step 15.3

Combine and .

Step 15.4

Multiply .

Step 15.4.1

Multiply by .

Step 15.4.2

Multiply by .

Step 16

Reorder terms.