Question

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

Answer 1

because the derivative of log x is 1/x?

Answer 2

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

Answer 3

But what's the reason? :(

Answer 4

We know that integration or anti-derivative is opposite of differentiation.

The differentiation can be proved by using various ways.

Answer 5

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 ?

Answer 6

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

Answer 7

and yes it's fascinating.

Answer 8

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

Answer 9

How can we prove integral of 1/x ?

Answer 10

If it's undefined then why write it ln ?

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

Lol.

Answer 15

fascinating*

Answer 16

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

Answer 17

If you find something else, let me know :D

Answer 18

Sure. ;)

Answer 19

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

Answer 20

agree with Fool

Answer 21

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

Answer 22

Awww. :(

Answer 23

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

Answer 24

Yes.

Answer 25

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

Answer 26

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

Answer 27

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?

Answer 28

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 ?

Answer 29

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

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

Might be..

Answer 34

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

Answer 35

is this what you are looking for?

Answer 36

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

Answer 37

@ninhi5 No. lol.

Answer 38

ooh sorry :(

Answer 39

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

Answer 40

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.

Answer 41

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

Answer 42

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)?

Answer 43

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

Answer 44

Lol. agree with the statement above^

Answer 45

@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)?

Answer 46

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

Answer 47

lol my bad typed to slow

Answer 48

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

Answer 49

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.

Answer 50

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.

Answer 51

@mathdood what did i miss?

Answer 52

is Purcel a gun or something? D:

Answer 53

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

Answer 54

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

Answer 55

Lol.

Answer 56

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/

Answer 57

THanks. i will try to read it. :D