# Why is the integral of 1/x ln x?

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We were doing FP1 topic on calculus and the teacher was about to do an example using 1/x but he was like oops it's an exception and the integral of 1/x is actually ln x.

I have no idea why this is true (my teacher didn't know/couldn't explain) and I have searched on Google but explanations there are incomprehensible for me.

Is there any way for me to understand this thing having only done C1 and half of FP1 so far? Will I be able to understand it by the end of A2 FM?

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We were doing FP1 topic on calculus and the teacher was about to do an example using 1/x but he was like oops it's an exception and the integral of 1/x is actually ln x.

I have no idea why this is true (my teacher didn't know/couldn't explain) and I have searched on Google but explanations there are incomprehensible for me.

Is there any way for me to understand this thing having only done C1 and half of FP1 so far? Will I be able to understand it by the end of A2 FM?

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#2

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Hello

We were doing FP1 topic on calculus and the teacher was about to do an example using 1/x but he was like oops it's an exception and the integral of 1/x is actually ln x.

**C0balt**)Hello

We were doing FP1 topic on calculus and the teacher was about to do an example using 1/x but he was like oops it's an exception and the integral of 1/x is actually ln x.

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#3

(Original post by

Hello

We were doing FP1 topic on calculus and the teacher was about to do an example using 1/x but he was like oops it's an exception and the integral of 1/x is actually ln x.

I have no idea why this is true (my teacher didn't know/couldn't explain) and I have searched on Google but explanations there are incomprehensible for me.

Is there any way for me to understand this thing having only done C1 and half of FP1 so far? Will I be able to understand it by the end of A2 FM?

Posted from TSR Mobile

**C0balt**)Hello

We were doing FP1 topic on calculus and the teacher was about to do an example using 1/x but he was like oops it's an exception and the integral of 1/x is actually ln x.

I have no idea why this is true (my teacher didn't know/couldn't explain) and I have searched on Google but explanations there are incomprehensible for me.

Is there any way for me to understand this thing having only done C1 and half of FP1 so far? Will I be able to understand it by the end of A2 FM?

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E.g. are you happy that the integral of x^2 is x^3/3? What explanation has made you accept this?

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#4

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You will learn about differentiating natural logs and exponentials in C3/4

**TenOfThem**)You will learn about differentiating natural logs and exponentials in C3/4

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#5

**C0balt**)

Hello

We were doing FP1 topic on calculus and the teacher was about to do an example using 1/x but he was like oops it's an exception and the integral of 1/x is actually ln x.

I have no idea why this is true (my teacher didn't know/couldn't explain) and I have searched on Google but explanations there are incomprehensible for me.

Is there any way for me to understand this thing having only done C1 and half of FP1 so far? Will I be able to understand it by the end of A2 FM?

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If we rearrange we see that now integrating both sides gives where c is constant but recall

so

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#6

(Original post by

E.g. are you happy that the integral of x^2 is x^3/3? What explanation has made you accept this?

**notnek**)E.g. are you happy that the integral of x^2 is x^3/3? What explanation has made you accept this?

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#7

(Original post by

But will you get told anything more than "the derivative of log x is 1/x but it's too difficult to explain why"? [Genuine question - I don't know how it's taught these days].

**DFranklin**)But will you get told anything more than "the derivative of log x is 1/x but it's too difficult to explain why"? [Genuine question - I don't know how it's taught these days].

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#8

At A-Level I was just told that it is what it is with no real reasoning.

I still don't really understand why it is but I'm starting a calc modulus after christmas and I know for a fact it's on there.

I still don't really understand why it is but I'm starting a calc modulus after christmas and I know for a fact it's on there.

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#9

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The most common way I've seen teachers/textbooks introduce it at C3 is by assuming the derivative of e^x and using y=ln(x) --> x=e^y with dx/dy = 1/(dy/dx).

**notnek**)The most common way I've seen teachers/textbooks introduce it at C3 is by assuming the derivative of e^x and using y=ln(x) --> x=e^y with dx/dy = 1/(dy/dx).

[Obviously, the involvement of 'e' is a massive spanner in the works at A-level, since it's a completely unfamiliar number and the definitions of it are pretty intractable ( is fairly easy to understand, but how the heck you work with the result is less so...)]

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#10

(Original post by

Hello

We were doing FP1 topic on calculus and the teacher was about to do an example using 1/x but he was like oops it's an exception and the integral of 1/x is actually ln x.

Is there any way for me to understand this thing having only done C1 and half of FP1 so far? Will I be able to understand it by the end of A2 FM?

**C0balt**)Hello

We were doing FP1 topic on calculus and the teacher was about to do an example using 1/x but he was like oops it's an exception and the integral of 1/x is actually ln x.

Is there any way for me to understand this thing having only done C1 and half of FP1 so far? Will I be able to understand it by the end of A2 FM?

To say that is the integral of is equivalent to saying that:

a) differentiates to

c) the integral behaves just like the log function

The most important property of the log function is that , so we want to show that we can make some integral have the same property (as well as the other properties of log, like and so on).

We can define a function as follows:

which computes the area under from 1 to some arbitrary value . This function is, in fact, the same as .

1. since the area under from 1 to 1 is 0

2. It should be more-or-less obvious that if we differentiate we get back to (since differentiating undoes integration)

3. so it behaves like a log function

That's because by splitting the area from 1 to into 2 chunks and adding them.

Now the first integral is simply by the defn of the function. Also if you make the substitution you can show that the second integral is simply as we wanted.

So has all the properties of , and hence *is* and also (obviously?) differentiates to .

That is a more advanced way to answer the question but not inaccessible to a reasonably bright further maths student, I think.

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#11

(Original post by

I don't think it's too hard to understand this.

To say that is the integral of is equivalent to saying that:

a) differentiates to

c) the integral behaves just like the log function

**atsruser**)I don't think it's too hard to understand this.

To say that is the integral of is equivalent to saying that:

a) differentiates to

c) the integral behaves just like the log function

**the**log function you talk about?"

Explicitly, you mean log to base e, but what is this mysterious 'e' and where does it come from?

Everything else you say is correct, but also correct if you define log(x) = for any alpha > 0.

At the end of the day, a lot depends on how you've defined e, I think.

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#12

(Original post by

And I assume you prove the derivative of e^x by using the derivative of ln x and using similar reasoning...

[Obviously, the involvement of 'e' is a massive spanner in the works at A-level, since it's a completely unfamiliar number and the definitions of it are pretty intractable ( is fairly easy to understand, but how the heck you work with the result is less so...)]

**DFranklin**)And I assume you prove the derivative of e^x by using the derivative of ln x and using similar reasoning...

[Obviously, the involvement of 'e' is a massive spanner in the works at A-level, since it's a completely unfamiliar number and the definitions of it are pretty intractable ( is fairly easy to understand, but how the heck you work with the result is less so...)]

You can get by in life/applied maths without knowing much more.

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#13

(Original post by

What is this "

Explicitly, you mean log to base e, but what is this mysterious 'e' and where does it come from?

Everything else you say is correct, but also correct if you define log(x) = for any alpha > 0.

At the end of the day, a lot depends on how you've defined e, I think.

**DFranklin**)What is this "

**the**log function you talk about?"Explicitly, you mean log to base e, but what is this mysterious 'e' and where does it come from?

Everything else you say is correct, but also correct if you define log(x) = for any alpha > 0.

At the end of the day, a lot depends on how you've defined e, I think.

However, my main point was to give someone a nudge towards seeing that there's a fairly obvious relationship between and log functions, and that by being aware that integration and differentiation are inverse operations to see that some log function should differentiate to .

As I'm sure you know, this can be made water-tight ("prove that there is a unique function L(x) with properties blah blah .. fundamental theorem of calculus blah blah .. unique inverse function E(L(x)) = x blah blah ..") but the details are out of place here.

(BTW, doesn't this discussion come round about every 2 months here, with the associated back-and-forth about chicken and egg definitions of e^x, e, ln(x), ooh that's a circular argument, no it isn't because I said such and such .. ?)

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#14

(Original post by

You're right, I jumped the gun a bit by equating the L function to the natural log, explicitly.

However, my main point was to give someone a nudge towards seeing that there's a fairly obvious relationship between and log functions, and that by being aware that integration and differentiation are inverse operations to see that some log function should differentiate to .

As I'm sure you know, this can be made water-tight ("prove that there is a unique function L(x) with properties blah blah .. fundamental theorem of calculus blah blah .. unique inverse function E(L(x)) = x blah blah ..") but the details are out of place here.

(BTW, doesn't this discussion come round about every 2 months here, with the associated back-and-forth about chicken and egg definitions of e^x, e, ln(x), ooh that's a circular argument, no it isn't because I said such and such .. ?)

**atsruser**)You're right, I jumped the gun a bit by equating the L function to the natural log, explicitly.

However, my main point was to give someone a nudge towards seeing that there's a fairly obvious relationship between and log functions, and that by being aware that integration and differentiation are inverse operations to see that some log function should differentiate to .

As I'm sure you know, this can be made water-tight ("prove that there is a unique function L(x) with properties blah blah .. fundamental theorem of calculus blah blah .. unique inverse function E(L(x)) = x blah blah ..") but the details are out of place here.

(BTW, doesn't this discussion come round about every 2 months here, with the associated back-and-forth about chicken and egg definitions of e^x, e, ln(x), ooh that's a circular argument, no it isn't because I said such and such .. ?)

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(Original post by

I'm not sure what kind of explanation you're looking for

E.g. are you happy that the integral of x^2 is x^3/3? What explanation has made you accept this?

**notnek**)I'm not sure what kind of explanation you're looking for

E.g. are you happy that the integral of x^2 is x^3/3? What explanation has made you accept this?

Yeah I guess so

I was told that integration is kind of inverse differentiation

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(Original post by

Ok here is a 'basic i.e. non-formal' idea: the derivative of is itself. i.e. let then

**tombayes**)Ok here is a 'basic i.e. non-formal' idea: the derivative of is itself. i.e. let then

I get the last part but it's meaningless because I don't get the very first bit lol

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#17

(Original post by

I don't get why this weird thing called e and log come into play suddenly at 1/x

Yeah I guess so

I was told that integration is kind of inverse differentiation

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**C0balt**)I don't get why this weird thing called e and log come into play suddenly at 1/x

Yeah I guess so

I was told that integration is kind of inverse differentiation

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But in the same way, can you look for something that differentiates to give 1/x? This is x^(-1) so using the same method as above, you may think that it's something involving x^0, but does this differentiate to give x^(-1)?

Since you haven't met logs or e yet, I'm not sure if you'll be able to get more of an explanation. I suggest either waiting until C3 or read a C3 textbook now / google it.

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#18

**C0balt**)

I don't get why this weird thing called e and log come into play suddenly at 1/x

Yeah I guess so

I was told that integration is kind of inverse differentiation

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#19

(Original post by

A Level starts by introducing e^x as a function with derivative e^x (no explanation but maybe an illustration using graphs of a^x functions and their gradient functions).

You can get by in life/applied maths without knowing much more.

**notnek**)A Level starts by introducing e^x as a function with derivative e^x (no explanation but maybe an illustration using graphs of a^x functions and their gradient functions).

You can get by in life/applied maths without knowing much more.

Then, as has been said, the differential of ln(x) follows

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**atsruser**)

I don't think it's too hard to understand this.

To say that is the integral of is equivalent to saying that:

a) differentiates to

c) the integral behaves just like the log function

We can define a function as follows:

which computes the area under from 1 to some arbitrary value . This function is, in fact, the same as .

1. since the area under from 1 to 1 is 0

2. It should be more-or-less obvious that if we differentiate we get back to (since differentiating undoes integration)

3. so it behaves like a log function

That's because by splitting the area from 1 to into 2 chunks and adding them.

Now the first integral is simply by the defn of the function. Also if you make the substitution you can show that the second integral is simply as we wanted.

So has all the properties of , and hence *is* and also (obviously?) differentiates to .

That is a more advanced way to answer the question but not inaccessible to a reasonably bright further maths student, I think.

which computes the area under from 1 to some arbitrary value . This function is, in fact, the same as .

1. since the area under from 1 to 1 is 0

2. It should be more-or-less obvious that if we differentiate we get back to (since differentiating undoes integration)

3. so it behaves like a log function

That's because by splitting the area from 1 to into 2 chunks and adding them.

Now the first integral is simply by the defn of the function. Also if you make the substitution you can show that the second integral is simply as we wanted.

So has all the properties of , and hence *is* and also (obviously?) differentiates to .

That is a more advanced way to answer the question but not inaccessible to a reasonably bright further maths student, I think.

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