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.

Add and .

Reorder and .

Using the Pythagorean Identity, rewrite as .

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 = .

Multiply by .

Multiply by .

Simplify.

Multiply and .

Multiply by .

Replace all occurrences of with .

Find the Integral sec(2x)^3