Name: Robert

Who is asking: Student
Level: Secondary

Question: How do you integrate secant(theta)?

I know the answer is ln [sec(theta) + tan(theta)] but how do you get there?

I am afraid that all methods involve pulling a rabbit out of a hat at some point. There is always a lot of guesswork involved in integration: Should you use integration by parts? If so, how? Or should you use substition? And if so, what should you substitute for what?... There is no precise procedure for answering these questions. Experience helps, and that is why you should practice, but you can always stumble on something new.

On the other hand, the fundamental theorem of calculus tells something really nice: Once you get the answer, then it does not make a difference whether you just guessed it or you worked for hours using intricate substitutions and many integrations by parts. You can always verify that your answer is the right one just by taking its derivative and ending up with what you started with. So, in this case, the simplest reason to explain that the answer is ln[sec(theta) + tan(theta)] is by taking its derivative and getting sec(theta) after simplification.

The "method" of integration I have seen in a book is to multiply sec(theta) by [sec(theta)+tan(theta)]/[sec(theta)+tan(theta)], and then substitute u for sec(theta)+tan(theta). But I don't know how you are supposed to guess that you should do that. Wouldn't it be simpler just to guess the answer?