# Calculus Examples

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

Simplify .

Simplify each term.

Apply the product rule to .

Raise to the power of .

Multiply by .

Factor out of .

Factor out of .

Factor out of .

Apply pythagorean identity.

Rewrite as .

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

Simplify.

Multiply by .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

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

Use the half-angle formula to rewrite as .

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

Combine and .

Split the single integral into multiple integrals.

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

Let . Find .

Rewrite.

Divide by .

Rewrite the problem using and .

Combine and .

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

The integral of with respect to is .

Simplify.

Replace all occurrences of with .

Replace all occurrences of with .

Replace all occurrences of with .

Combine and .

Apply the distributive property.

Combine and .

Multiply .

Multiply and .

Multiply by .

Reorder terms.