# Integrating 1/x

Yeah, this questions may be a little elementary for some, but I don't seem to have any sources which would be able to tell me how do i integrate a function 1/x. Any help would be great.

: )

## Answers and Replies

arildno
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Gold Member
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I assume that you know that an antiderivative is ln(x), but you don't know how to get there?

arildno said:
I assume that you know that an antiderivative is ln(x), but you don't know how to get there?
whoa crap, ln (x) didn't cross my mind. Thanks alot........

ln(x) will never "cross your mind." You have to either memorize it or know how to derive it.

$$x=e^y$$
$$\frac{dx}{dy}=e^y=x$$
$$(1/x)dx=dy$$
$$\int (1/x) dx=\int dy = y+constant = lnx+constant$$

StatusX
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$$\int_1^x t^{-1 + \epsilon} dt$$

For $$\epsilon$$ greater than 0. Obviously this would be:

$$\frac{x^{\epsilon} - 1}{\epsilon}$$

And you can see by applying l'hopitals that this goes to ln(x) as $$\epsilon$$ goes to 0.

Now if you want to get more technical, and not assume anything, you can see that the derivative of a^x is:

$$\lim_{h \rightarrow 0} \frac{a^{x+h} - a^x}{h} = a^x \lim_{h \rightarrow 0} \frac{a^h - 1}{h}$$

Now if the limit exists, which you can see it clearly does by looking at the graph, this means that the derivative is some function of a times a^x. You can easily check that for a less than one, this function of a is negative, its equal to 0 at one, and its greater than 0 for a greater than one, and continues getting bigger and bigger. For some a, it will be exactly equal to one, and we'll call this e. So the derivative of e^x is just e^x. The inverse of exponentiation is logarithm, and ln(x) is defined as the log base e. Then the derivative of a^x is the derivative of e^(ln(a)*x) which you can see from the chain rule is ln(a)*e^(ln(a)*x), or just ln(a)*a^x. So the function of a was just ln(a). This can be used in the limit above:

$$\lim_{\epsilon \rightarrow 0} \frac{x^{\epsilon} - 1}{\epsilon} = \lim_{\epsilon \rightarrow 0} \frac{ln(x) \cdot x^{\epsilon}}{1} = ln(x)$$

Actually, you don't even have to use l'hopitals at this point, because if you look carefully you'll see this limit and the funtion of a are indentical, just with a replaced by x and h replaced by $$\epsilon$$.

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StatusX
Homework Helper
Actually, you can take this even farther and use it to find the value of e:

$$\ln(x) = \lim_{\epsilon \rightarrow 0} \frac{x^{\epsilon} - 1}{\epsilon}$$

Now noting that e^x is the inverse of ln(x):

$$y = e^x$$

$$x = \ln(y) = \lim_{\epsilon \rightarrow 0} \frac{y^{\epsilon} - 1}{\epsilon}$$

rearranging (and noting that these expressions are only valid in the limit as $$\epsilon$$ goes to 0):

$$\epsilon \cdot x = y^\epsilon - 1$$

$$y^\epsilon = 1 + x \cdot \epsilon$$

$$y = (1 + \epsilon \cdot x)^{1/\epsilon}$$

which leaves:

$$e^x =\lim_{\epsilon \rightarrow 0} (1 + \epsilon \cdot x)^{1/\epsilon}$$

or, plugging in 1 for x, and replacing $$\epsilon$$ with 1/n:

$$e = \lim_{n \rightarrow \infty} (1 + 1/n)^n$$

Which is the standard way of defining e. I'm sorry if this is much more than you wanted. I never really thought about this stuff, just accepting it at face value, and your question prompted me to try to derive it myself. I was suprised how nicely it all works out.

misogynisticfeminist said:
whoa crap, ln (x) didn't cross my mind. Thanks alot........
Don't forget that in calculus they use log[x] as the natural log, instead of ln[x].

dav2008
Gold Member
Chrono said:
Don't forget that in calculus they use log[x] as the natural log, instead of ln[x].
Can you explain that...?

Hey, thanks for showing how the integral for 1/y was obtained, that was helpful....

: )

dav2008 said:
Can you explain that...?
Pretty much every calculus book uses log[x] instead of ln[x] to represent the natural log with base e. Remeber that in high school they told you that if you had log[x] without a base it was understood to be 10? Well, that's not ture in calculus. Log[x] has a base e. Is that better?

dav2008
Gold Member
Chrono said:
Pretty much every calculus book uses log[x] instead of ln[x] to represent the natural log with base e. Remeber that in high school they told you that if you had log[x] without a base it was understood to be 10? Well, that's not ture in calculus. Log[x] has a base e. Is that better?
I have never seen log(x) used as the natural log

dav2008 said:
I have never seen log(x) used as the natural log
I see it used all the time. Even Mathematica uses it like that.

Yes, but only mathematicians.

I don't go near engineers, because they smell of booze and modernism, but I'm told that they use log to mean log to the base 10.

Galileo
Homework Helper
It's just a notational convention, not a 'new' definition for the notation log(x) because it's calculus. In my book on statistical mechanics they also use log for ln.
The author writes:
Note: All logarithms in this book are natural logarithms - The base 10 logarithm is as much of a historical curiosity as a slide rule.)
My (university) calculus book uses ln, for log base e. Simply to seperate it from the other bases.

Yes, but only mathematicians.

I don't go near engineers, because they smell of booze and modernism, but I'm told that they use log to mean log to the base 10.
My statistical mechanics book (Kittel) also uses log to mean ln. It has something to do with clarity. ln can be confused easily as a product of l and n or something...

BobG
Homework Helper
Note: All logarithms in this book are natural logarithms - The base 10 logarithm is as much of a historical curiosity as a slide rule.)
Base 10 log is used to measure gain in power amplification, used to measure sound levels, used to measure earthquakes, used to measure power of radio signals and gain of antennas among other things.

Even more importantly,
What do they mean slide rules are only a historical curiosity!! I still use mine!

:grumpy:

And what's all this talk of smelling like booze Geez, I've gotta find a stronger aftershave

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matt grime
Homework Helper
But as log to any base only differs by a multiplicative constant, it is merely a change of unit away from being log in any other base.

log_{10} is useful since it is, give or take a small number, the number of digits in the original number. but it's still only factor of 2 and a bit away from log proper....

as we were told:

log will always mean log base e from now on. After all, why on earth would you want to use anything else?

why would log mean log_e ?? thats what we have ln for !! Everything I've learned says that ln is log_e and log is log_10.
As for the integration part you just have to remember that any equation with 1/x or x^-1 will have a ln in it's answer.
I alway remember by doing this problem.
what is the antiderivative of 1/chair ?
I'll let you figure it out.

Galileo
Homework Helper
BobG said:
Base 10 log is used to measure gain in power amplification, used to measure sound levels, used to measure earthquakes, used to measure power of radio signals and gain of antennas among other things.
Which is actually curious. They should've used the natural logarithms.
Who was this dopey guy who invented this dB scale? I think it was Alexander Graham Bell, who invented the telephone.
Well, that means he was an engineer so that figures :rofl:

...I was just kidding BobG, no need to get your underwear into a wrinkle.

I've gotten used to using lg as log_{10} and ln as log_{e}.

I just think it's safe to always assume log[x] as being the natural log, unless otherwise specified.

vsage
I think engineers are right in this case. Ln is the notation for log base e in other countries (and not log), so I am told by my spanish (from spain) linear alg. prof. They don't call it the natural log though, they call it the Naperian log. That's curious because Napier lived well before Euler ever conceived of e.

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Alkatran