# Calculus Examples

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

Rewrite as .

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

Multiply by to get .

Multiply by to get .

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Remove parentheses.

Simplify each term.

Multiply by to get .

Multiply by to get .

Multiply by to get .

Simplify .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and to get .

Add and to get .

Add and to get .

Apply pythagorean identity.

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

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and to get .

Use the half-angle formula to rewrite as .

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

Since integration is linear, the integral of with respect to is .

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

Let . Then , so . Rewrite using and .

Write as a fraction with denominator .

Multiply and to get .

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

The integral of with respect to is .

Simplify.

Replace all occurrences of with .

Replace all occurrences of with .

Replace all occurrences of with .

Simplify each term.

Simplify each term.

Divide by to get .

Divide by to get .

Apply the distributive property.

Simplify .

Write as a fraction with denominator .

Multiply and to get .

Simplify .

Multiply and to get .

Multiply by to get .

Reorder terms.