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.
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Multiply by .
Multiply by .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply .
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Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Add and .
Add and .
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 .
Use the half-angle formula to rewrite as .
Since is constant with respect to , move out of the integral.
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
Let . Then , so . Rewrite using and .
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Let . Find .
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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 .
Simplify the answer.
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Simplify each term.
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Divide by .
Divide by .
Combine and .
Apply the distributive property.
Combine and .
Multiply .
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Multiply and .
Multiply by .
Reorder terms.
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