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 .
Simplify.
Multiply by to get .
Multiply by to get .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Remove parentheses.
Simplify and combine like terms.
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.
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.
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 .
Combine fractions.
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.