# Find the Integral sec(2x)^3 Let . Then , so . Rewrite using and .
Let . Find .
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.
Reorder and .
Using the Pythagorean Identity, rewrite as .
Simplify by multiplying through.
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.
Raise to the power of .
Use the power rule to combine exponents.
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.
Multiply by .
Multiply by .
Simplify.
Simplify.
Multiply and .
Multiply by .
Replace all occurrences of with .
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