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 .
Expand using the FOIL Method.
Simplify and combine like terms.
Simplify with factoring out.
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Factor out of .
Factor out of .
Factor out of .
Apply pythagorean identity.
Simplify with factoring out.
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Rewrite as .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Since is constant with respect to , the integral of with respect to is .
Use the half-angle formula to rewrite as .
Since is constant with respect to , the integral of with respect to is .
Combine fractions.
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Write as a fraction with denominator .
Multiply and .
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 .
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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Apply the distributive property.
Simplify .
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Write as a fraction with denominator .
Multiply and .
Simplify .
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Multiply and .
Multiply by .
Reorder terms.
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