# Calculus Examples

Step 1
Let , where . Then . Note that since , is positive.
Step 2
Simplify terms.
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
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
Let . Then , so . Rewrite using and .
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
Substitute back in for each integration substitution variable.
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
Simplify.
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.