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 to get .
Multiply by to get .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Remove parentheses.
Simplify and combine like terms.
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Simplify each term.
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Multiply by to get .
Multiply by to get .
Multiply by to get .
Simplify .
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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 .
Combine fractions.
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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 the answer.
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Simplify.
Replace all occurrences of with .
Replace all occurrences of with .
Replace all occurrences of with .
Simplify each term.
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Simplify each term.
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Divide by to get .
Divide by to get .
Apply the distributive property.
Simplify .
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Write as a fraction with denominator .
Multiply and to get .
Simplify .
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Multiply and to get .
Multiply by to get .
Reorder terms.
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