Find the Integral sec(2x)^3

Math
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.
Factor out of .
Integrate by parts using the formula , where and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Simplify the expression.
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Add and .
Reorder and .
Using the Pythagorean Identity, rewrite as .
Simplify by multiplying through.
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Rewrite the exponentiation as a product.
Apply the distributive property.
Reorder and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
The integral of with respect to is .
Apply the distributive property.
Solving for , we find that = .
Simplify.
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Multiply by .
Multiply by .
Simplify.
Simplify.
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Multiply and .
Multiply by .
Replace all occurrences of with .
Find the Integral sec(2x)^3

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