View Full Version : integral of 1/x

Silvercord

03-14-2004, 02:11 PM

yes this is probably a really stupid question, but 1/x doesn't have an integral right?

I wrote a scripting language to graph the areas under the curve (using a numerical approximation algorithm) and on any interval with -x and x (the absolute value of x is the same) I always get an area under the curve of zero, which is what it *seems* like it should be if you consider negative area (then they just cancel), but I can't exactly prove it.

Also, I'm not in college, so if you DO have some fancy proof I probably wont' understand it anyway because i suck

EDIT:

the only thing I can think of is to say that the functions are changing at the same rate when they are the same distance from zero, and that might satisfy that they are approaching the 'same infinity' (i.e i guess you cannot prove that the tangent of x from 0 to PI is approaching the 'same infinity' at pi/2, my math teacher said that one could 'approach infinity before the other' in which case the area is either positive or negative infinity but not zero).

well I wont prove it for you, but the integral of 1/x is ln(|x|)

this .pdf might be handy: http://www.riverdeep.net/math/tangible_math/tm_handouts/investig/fi47.pdf

Silvercord

03-14-2004, 02:39 PM

hmm, as soon as I read that I remembered that the derivative of ln(x) is equal to 1/x but I've never read a proof.

That looks like a good link. I'm just thinking right now.

Thanks :)

Silvercord

03-14-2004, 02:53 PM

wait, something else I want to know, what is ln(x) limited by as x approaches infinity? I don't know if I'm already supposed to know this or what.

EDIT:

I guess it's limited by infinity?!¿ That's weird because it's getting slower and slower, but isn't limited

right limit of ln( x ) as x approaches infinity is infinity

and lim x->0+ of ln(x) = -∞

edit:: thats why the range of of ln(x) is (-∞, ∞)

bludstayne

03-14-2004, 07:30 PM

According to integrals.wolfram.com, the integral of 1/x is Log[X]

neandrake

03-14-2004, 09:26 PM

blah and meh

Originally posted by neandrake

blah and meh

thanks a lot, captain obvious

undisputed007

03-16-2004, 08:55 AM

Originally posted by bludstayne

According to integrals.wolfram.com, the integral of 1/x is Log[X]

according to my studies the integral of 1/x is ln(x)

not log(x)

Govtcheez

03-16-2004, 09:14 AM

Originally posted by axon

same thing :rolleyes:

Are you sure? log and ln have different bases, unless he's got a weird math book.

Thantos

03-16-2004, 09:18 AM

Are you sure? log and ln have different bases, unless he's got a weird math booQuite a few of the math papers i've read define Log to be LN. If they want log10 they actually write log10.

Not sure of the reasonning but I've confirmed with several of the math professors that it is a common practice.

EvBladeRunnervE

03-16-2004, 09:33 AM

. If they want log10 they actually write log10.

Yeah, I have seen this quite a bit also. I believe it has to deal witht the fact that LN is the most commonly used log, as all other logs can be derived from it.

>> I believe it has to deal witht the fact that LN is the most commonly used log, as all other logs can be derived from it.

I think you hit the problem right on the head! since log could be of any base (mainly 2 or 10), we don't use it....but since ln is always of base e it could be use to calculate every logarithmic table.

Silvercord

03-16-2004, 11:35 AM

can you take the derivative using the log of any base and not just ln? i'm reading about 'logarithmic differentiation' and so far I only have yet to see them taking the natural log of both sides of the equation to solve for the derivative of some function

golfinguy4

03-16-2004, 02:45 PM

Yep, you can. When you take the derivative, there will be a ln 10 on both sides which cancel out through division.

Silvercord

03-17-2004, 02:55 PM

do any of you guys know of the scientific way to compute the logarithm of a number. I want to program my own natural logarithm function, just for the heck of it, which I guess I could then in turn use for any other base by using change of base formulas.

EvBladeRunnervE

03-17-2004, 03:09 PM

from mathforum.org:

ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...(repeating to infinity)

mathforum (http://mathforum.org/library/drmath/view/55540.html)

Silvercord

03-17-2004, 03:18 PM

sweet dude thanks

EvBladeRunnervE

03-17-2004, 03:31 PM

no problem, I started calculus I this semester, but have read a head a bit, so I knew alittle bit more what to look for(for instance, I knew to look for stuff with infinite series).

Silvercord

03-17-2004, 03:41 PM

that's pretty cool. we might be on the same level, where exactly are you at? I'm in chapter 7 of our book, which is about half way (we're starting integrals). If we're at exactly the same spot we might be able to volley questions back and forth when we are doing homework and stuck. I really like calculus I'm getting a retardedly high grade in it and I *think* I understand everything at an intuitive level. want to read my latest portfolio? it's about the fundamental theorem of calculus explaining why and how integrals work...sweet stuff d00d!!!!!!!!!!!

EvBladeRunnervE

03-17-2004, 05:04 PM

Well, we have just gotten past the part in our covering derivatives, and all of the shortcuts to differentiation and we now are working on the application side of derivatives, we touch integrals in a little bit though.

I am ........ed off at myself though right now, I think botched a test I should of aced because of a stupid stomach bug that I got at the same time as one of those "my head is so cloudy I can barely think" cold bugs(also, I attributed a principle of a derivative of a function to the function itself, the " there is no derivative at a point, if the point is a sharp corner of the function" property of derivatives... I attributed it to continuous functions in general....(another reason why poor sleeping habits suck in college).

It is great fun though, what is your text book, by any chance? and can you give me a basic overview of what you have covered so far?.

EDIT: In case your wondering about how well I do in calculus, besides messing up on the test, I have a score of 99% in the class.

Silvercord

03-17-2004, 05:46 PM

are you talking about a function such as:

y = abs(x)

where x is zero and has no derivative?

My book is called "Calculus: Algebraic and Numerical"...I don't remember off hand I can get the exact name and authors.

We've covered all of the basic calc stuff (limits and continuity and derivatives). The hardest stuff we've done was related rates. that's the same score i got first semester!

EvBladeRunnervE

03-17-2004, 05:49 PM

wait a sec, your second semester in calculus?

where x is zero and has no derivative?

yeah, y = abs(x) is the type of function I am talking about. I really dislike messing up on tests(hell, I get royally ........ed if I score below 95 on anything scientific).

Silvercord

03-17-2004, 05:53 PM

ya but my class is really slow...we're collectively retarded, im the only one who 'gets' any of it. I said i'm in chapter 7, but the class is in chapter 5. Were *just* starting integrals, so I'm sure you and I are pretty damn close to being at the same level.

EvBladeRunnervE

03-17-2004, 05:59 PM

Yeah, the class goes pretty slow for me to, I am planning on reading ahead and getting done with all the calc 2 stuff during the summer(single variable integration, the whole works). I honestly dont understand why people have the collective stupidity when it comes to derivatives, what is so hard to understand about instantaneous rate-of-change? Alot of people in my class have problems with the shortcuts to differentiation, because they never tried solving for it using the old limit as h -> 0 of (f(x+h) - f(x))/h , well, it is because they dont know how to generalize things, for example, the chain rule can be generalized fromthe derivative of e^(u(x)).

Silvercord

03-17-2004, 06:08 PM

no no no using that is the worst way to teach the chain rule...the chain rule is the easiest and simplest calculus idea ever so you shouldn't obfusc it by trying to use e^u caca

EvBladeRunnervE

03-17-2004, 06:11 PM

I know it is simple, however, noone really tries to figure it out by themselves the hardest way possible(i.e. the way that usually you will get the most from it if you understand it at all).

EDIT: Sort of like finding the derivative of sin x. you get to a point where it is : sin x lim [(cos h - 1)/h] + cos x lim (sin h)/h. We know the limit of sin h / h is 1 and the limit of cos h -1 / h is 0.

Silvercord

03-17-2004, 08:43 PM

Why would you want to figure out the chain rule the hardest way possible? It's the easiest thing to get an intuitive understanding of. I can't remember how to derive the sandwich rule or the trig identities unfortunately :(

Ill have to read up on that again

EvBladeRunnervE

03-17-2004, 08:58 PM

trig identities go as such,

remember that f' = [f(x+h) - f(x)]/h as h ->0

sin(x+h) = sin x cos h + cos x sin h

therefore:

f' = lim (sin x cos h + cos x sin h - sin x )/h as h ->0

using the properties of limits:

f' = lim (sin x cos h - sin x)/h + lim cos x sin h/h as h->0

factor sin x:

lim sin x(cos h - 1)/h + lim cos x sin h/h as h->0

based on the fact that sin x and cos x have nothing to do with h(i.e. they are not involved in the limit):

sin x lim (cos h-1)/h + cos x lim sin h/h as h->0

computing the limit of (cos h - 1)/h and sin h/h as h -> 0 we have

sin x * 0 + cos x * 1 = cos x

EDIT: I know you can find out for yourself, just wanted to save you some trouble :P

and about chain rule, you are right, it is pretty intuitive:

if y = f(x)

and z = g(x)

and h(x) = f(g(x))

dy/dx = dy/dz * dz/dt

It is just that doing it this way tends to make people think of leibnix notation as a fraction, which in some ways it is, but most of the time it is better to think of it as just a notation.

Zach L.

03-17-2004, 09:00 PM

Sandwich rule?

EvBladeRunnervE

03-17-2004, 09:05 PM

Sandwich rule

Sandwiching is the following, say if I have 3 functions:

f(g(h(x)))

the derivative of this is : f'(g(h(x))) * g'(h(x)) * h'(x)

Zach L.

03-17-2004, 10:22 PM

Right... I can't say I had ever heard that term before.

lmao...I've never heard of it either...made me hungry though

DavidP

03-18-2004, 12:34 AM

Sandwiching is the following, say if I have 3 functions:

f(g(h(x)))

the derivative of this is : f'(g(h(x))) * g'(h(x)) * h'(x)

ive never heard that called the Sandwich rule...i usually here it called the Chain Rule.

EvBladeRunnervE

03-18-2004, 09:19 AM

I know, but I was just making a guess at it, as I honestly dont know what it is either....

Silvercord

03-18-2004, 10:25 AM

sandwich rule == sandwich theorem

Maybe you've heard of it differently but it's where:

sin(h) / h == 1

lim h -> 0

undisputed007

03-18-2004, 12:43 PM

Originally posted by Silvercord

sandwich rule == sandwich theorem

Maybe you've heard of it differently but it's where:

sin(h) / h == 1

lim h -> 0

Is that what you call sandwich rule??

I think i took it in Calculus but i ve never heard the word sandwich when i studied it

to be frank man the law you ve stated is derived from L'Hopital rule because if you substitute both sin(h) and h with zeros you will get 0/0

so in Hopital rule u differentiate both numerator and denominator by h so it will be cos(h)/1 where cos(0)=1

thus sin(h)/h =1

so it's L'hopital rule not sandwich one :)

Silvercord

03-18-2004, 08:36 PM

no i re-read it in my book im right

golfinguy4

03-18-2004, 10:22 PM

Originally posted by Silvercord

no i re-read it in my book im right

No you aren't. What book are you reading? The only "Sandwich Rule" I've heard of is used to describe limits of a function by "sandwiching" it between two functions.

EDIT: The Squeezing (Sandwich) Theorem (http://mathworld.wolfram.com/SqueezingTheorem.html)

The Actual Sandwich Theorem (http://mathworld.wolfram.com/SandwichTheorem.html)

jverkoey

03-18-2004, 10:26 PM

i don't know too much about calc, but i found a site on this theorem, if it helps any...:

http://www.sosmath.com/calculus/limcon/limcon03/limcon03.html

Silvercord

03-19-2004, 07:07 AM

No you aren't. What book are you reading? The only "Sandwich Rule" I've heard of is used to describe limits of a function by "sandwiching" it between two functions.

well maybe it's not the only example of the sandwich/pinching theorem but i'm on page 127 of "Calculus, Graphical, Numerical, Algebraic" by "Finney, Thomas, Demana, Waits" and it uses sin(h) / h, and that's the only way I had ever seen it done until you posted your links.

EvBladeRunnervE

03-19-2004, 07:47 AM

thanks jverkoey, that stuff is actually in my calculus book. wonder why I never realized it.

Powered by vBulletin® Version 4.2.5 Copyright © 2019 vBulletin Solutions Inc. All rights reserved.