Throughout my course this year ive had close to zero trouble on anything. But any time some integrals or derivatives of natural logs show up it just blows my mind.

examples:

(integral) lnx/x = 1/2(lnx)^2
(integral) lnx/x^2 = -lnx/x
(integral) cscx = ln(abs)(cscx-cotx)

things like that. especially the last one. the integrals of trig functions ending up with the ln of something confuse the hell out of me.

If anyone could explain this to me it would be greatly appreciated

Edit: and also the process behind solving the integrals

This post was edited by tidus_ffx2 on Oct 26 2013 04:57pm

for the first one :

let u = ln(x)
du = dx/x

substitute --> integral(u*du) --> u^2/2 where u is ln(x)


the second one is by parts



last one is some trigonometric manipulations : http://www.math.com/tables/integrals/more/csc.htm

the secret is just to learn them..its gonna be a pain in the ass if you demonstrate it everytime you wanna use them. the same goes for a lof of integrals, such as xln(x) etc

rewriting the integrand:

integral of ln(x) * (1/x) * dx

What is the derivative of ln(x)? It is (1/x), you should have this memorized. Now look at the integrand. We ln(x), and its derivative (1/x). This screams U substitution.

u = ln(x)
du/dx = 1/x
du = (1/x) dx
x*du = dx

now make the substitutions:

integral u * (1/x) * x * du
look the x's cancel!
so rewriting it:
integral u*du

which is just u^2/2, or (1/2)* u^2
now just reverse the substitution and you get
(1/2) * ln(x)^2

/e +C

This post was edited by Azrad on Oct 26 2013 05:08pm

Thanks guys. Ofc i know the derivative of lnx=1/x

So im guessing for the trig ones like #3 its something i should memorize?

the key for those is knowing your Trig Identities. Schools just don't seem to teach much Trig anymore, so this trips up many Calc students

This seems to be my issue actually. I know a few of them, but i suppose i should start learning the rest

Oh sorry, yeah don't take it personal, I just wanted to be clear (in case you didn't know, I figured you must know it).

When I was in college I carried a table of integrals which had an entry for all 6 trig functions (and much more of course).

If you do not have this luxury, and you haven't memorized them, then you must do them yourself. The idea is very similar to the first one I explain. You need to manipulate it to the point where you have a value times its derivative. Like in the example I did above you had ln(x) * (1/x)... Exactly how to do this with this problem is not obvious, imo. The process is described below. If you couldn't see this, I wouldn't be concerned, I couldn't see it either.
http://www.math.com/tables/integrals/more/csc.htm

This post was edited by Azrad on Oct 26 2013 05:46pm

Well one of the first things we did in class was memorize all trig derivates/ inverse trig/ hyperbolic/ inverse hyperbolic. But he never had us memorize the indefinite integrals, so now its just messin with me.

Appreciate the help though

Surely i will be studying these trig identities more and i suppose now that i know how to do it without memorizing it i could do it regardless. The proof of it was not hard, just something i havent seen before. Our professor likes to leave out huge details in class for us to figure out ourselves.

This post was edited by tidus_ffx2 on Oct 26 2013 05:58pm