So i was wondering, why the integral of 1/x is log x? Any particular reason?

because the derivative of log x is 1/x?

Yes since the differentiation of \(\ln x\) is \(\frac 1x \)

But what's the reason? :(

We know that integration or anti-derivative is opposite of differentiation. The differentiation can be proved by using various ways.

Yes. i know BUT Why integral of 1/x is ln ? why not something else.. Just like, integral of 1/x^2 = -1/x + C ?

Because the differentiation of 1/x is -1/x^2

and yes it's fascinating.

because if you use power rule on good old 1/x it would be undefined

How can we prove integral of 1/x ?

If it's undefined then why write it ln ?

I am not aware of any other definition of the indefinite integration other than anti-derivative approach. If it was definite then I would suggest Riemann's sum

i want to know the logic behind 1/x = ln x?

There is none other than what I had already said. It's fascination like \( \pi\) keeps poping in various places in mathematics.

Lol.

fascinating*

@asnaseer @dpaInc , do you guys know any other logic?

If you find something else, let me know :D

Sure. ;)

get the integral of \(\int \frac{x}{x^2} dx\)

agree with Fool

\[\int\limits \frac{\cancel x }{x^{\cancel2}}\to \frac 1x\]

Awww. :(

if i'm not mistaken some books actually define. lnx as the integral of 1/x

Yes.

\[\int x(x^{-2}) dx\] use product rule

is there any reason/logic mention behind? :D @dpaInc

the area under the curve 1/x from 1 to x is lnx that's what i was referring to... yes, some define lnx as that... how u gonna go against a definition?

umm, i never studied the definition. The part which confuses me is, why can't we use this thing: http://www-rohan.sdsu.edu/~jmahaffy/courses/f00/math122/lectures/de_int/images/deint4.gif in the case of 1/x ?

you got me there... i'd have to review this... :)

That holds for all \( n \in \mathbb{N} \)

1/x^2 = x^-2 so we can't use it here? D: @FoolForMath

maybe it's that reason they decided to define it that way so to avoid x^0 ? nah... just thinking outta my...

Might be..

\[d/dx \ln(x) \lim(d->0) (\ln(x+d)-lnx)/d\]

is this what you are looking for?

Rephrasing: \(\forall n \in \{\mathbb{Z} -\{-1\}\} \)

@ninhi5 No. lol.

ooh sorry :(

@ninhi5 , im looking for a logic behind: integral of 1/x = ln x?

like FFM, the only reason I can think of uses the fact that the derivative of ln(x) is 1/x, and integral is opposite of differentiation.

Yes. that's what i learnt. BUT i was not told the reason behind it. :(

otherwise why not argue this question: why is integral of x^n = x^(n+1)/(n+1) if integral of 1/x is ln(x)?

haha... good point... ^^^

Lol. agree with the statement above^

@saifoo.khan Well he was giving you the start of the proof for the derivative of ln(x) is 1/x would proving that the derivative of ln(x) is 1/x be enough to show that the integral of 1/x is ln(x)?

Who he? :D there are so many people answering.

lol my bad typed to slow

don't make me take out Purcell from the shelf!

ninhi was the he or she haha either way it seems someone else already made this argument and persuaded you so way to slow lol.

from: http://en.wikipedia.org/wiki/Integral under section "Fundamental theorem of calculus" The fundamental theorem of calculus is the statement that differentiation and integration are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved. An important consequence, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.

@mathdood what did i miss?

is Purcel a gun or something? D:

I totally agree that integration an differentiation are totally opposite of each other. But we can show the working for finding the integral of 1/x^2 Why is there no working for 1/x That's the main question! :D

lol... that WAS the defacto calculus text in colleges about ....say, couple hundred years ago

Lol.

there is an interesting analysis of this here: http://arcsecond.wordpress.com/2011/12/17/why-is-the-integral-of-1x-equal-to-the-natural-logarithm-of-x/

THanks. i will try to read it. :D

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