Calculus Examples

Step 1
Let , where . Then . Note that since , is positive.
Step 2
Simplify terms.
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Simplify .
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Simplify each term.
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Apply the product rule to .
Raise to the power of .
Multiply by .
Factor out of .
Factor out of .
Factor out of .
Apply pythagorean identity.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Simplify.
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Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
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 .
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Let . Find .
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Differentiate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
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.
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Replace all occurrences of with .
Replace all occurrences of with .
Replace all occurrences of with .
Step 15
Simplify.
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Combine and .
Apply the distributive property.
Combine and .
Multiply .
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Multiply by .
Multiply by .
Step 16
Reorder terms.
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