Page 29 - Calculus for the Clueless

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You might try to show that the answers to 9A and 9B are the same using the identity sec x = tan x + 1.
Example 10—

This is the worst case: m, the power of tan x, is even— m = 0—and n, the power of the sec x, is odd—n = 3. All
cases where m is even and n is odd are done by integrating by parts. They get long fast as the powers of m and n
increase, and all involve the same tricks.

Solving for the unknown integral, we get

Example 11—

Anytime you have a mixed integral, that is, where the tan is not with the sec, you will have to use trig identities
and usually tricks and sometimes long problems involving techniques that may not have been done here yet.
The one I've given is a rather mild one.