Indefinite Integrals: How Were They Figured Out?

[jdavel]

I've been wondering how all those indefinite integrals in a comprehensive table were figured out. Can they all be done with one (or some combination) of the standard methods, (substitution, parts etc.)? Or did somebody just poke around until they figured them out? For example, how do you find that Int(dx/cosx) = ln(1/cosx + tanx)?

 

[Gib Z]

All the indefinite integrals is tables can either be done by some integration method, and not just differentiating a lot of functions to see if they give what they want. Its not always a standard method though, some tricks are sometimes required. Other times, there actually is no elementary anti-derivative and they instead define that integral to be another function.

For your specific integral, There are a few ways of doing it.

A common way is (writing 1/cos x as sec x) multiplying the integrand through by (sec x + tan x). However, that makes it seem like you've already done this before and hence you know you can rely of this otherwise remarkable step.

So the way I prefer to do it as many people might see more easily, though it takes some more work. Multiply the integrand through by cos x, use the pythagorean identity on the denominator, a simple substitution and partial fractions, were home free =]

 

[jdavel]

Gib Z,

Very nice!

Let me ask you something. If you were teaching integration, how would you explain to your students what went through your head to come up with the idea of multiplying the integrand by cos(x)/cos(x)? Is there an insight that could be used when they hit another integral that doesn't seem to have an obvious method of solution? Did you see all at once the whole "...cos squared of x in the denominaor is going to give me a function of sin(x) through the Pythagorean theorem, and the differential for that will have cos(x) in it, which is just what I'll need for the cos(x)dx that I've created in the numerator..." ?

 

[Gib Z]

Its all about experience and experimenting. Even with all the standard methods of integration, there's always certain things you pick you from your own experience about what will work and what won't. Eventually you get to a point where..well yes, you do see the whole quote straight away.