Indefinite integrals - I'm really stuck

[xailer]

hello

I started learning indefinite integrals and already there are few things confusing me.

I understand that given a function f(x) an anti-derivative of f(x) is any function F(x) such that F'(x) = f(x). There are actually an infinite number of functions that we could use and they will all differ by a constant C, so

(F(x) + C)' = F'(x) = f(x) .

But few thigs I don't understand when writting down the indefinite integral using the symbol # (I will use # since I don't know how else to display the symbol ):

#f(x)dx = F(x) + C

What is dx doing in above formula ? Doesn't f(x)*dx give us differentials:

dF = f(x) * dx

, where dF is small change in function F and dx small change in variable x?
What on earth has this got to do with definition of integral I gave at the beginning of the post?




Since I guess terminology is giving me some troubles :

*What does the word differentiating mean ( it seems it has different meanings depending on the context we use it in )?

thank you

 

[pka]

You have asked a really good question about the dx.
First I want to warn you not to conflate the idea of ‘Integral’ and ‘antiderivative’.
While they are related they are most definitely not the same.
In higher mathematics an integral is always a number.
In fact I studied integration theory with someone who made all his students write \(\displaystyle \int {f(x)}\) without the dx.
When I began teaching calculus it took me for ever to get in the habit of writing \(\displaystyle \int {f(x)} dx\).
Even then, I explained that the dx is there to tell us the variable of integration.
When you study the numerical integration, Riemann Integral, you will see the use of \(\displaystyle \Delta x\) in the approximating sums. That symbol became the dx we find in textbooks.

In probability we might have this integral, \(\displaystyle \int_0^{.5} {\left( {yx - x^2 } \right)dx}\). The dx the tells us that we find the antiderivative with respect to x, \(\displaystyle \{\left. {(1/2)yx^2 - (1/3)x^3 \right|\nolimits_{x = 0}^{x = .5}\).

 

[ChaoticLlama]

You are correct in the statement that 'dx' is used for differentials. However, when used in integrals it still means 'a very small change in x.

To explain this better, I shall give the definition of integrals.

\(\displaystyle {\lim }\limits_{||p|| - > 0} \sum\limits_{k = 1}^n {f(x_k )\Delta x_k = \int\limits_a^b {f(x)dx} }\)

Notice the similarities between both side of the equation.
The symbol ||P|| represents the norm of the partition. The norm is the largest division that has been made when subdividing the function into rectangles. (remember that finding the area under a curve is technically summing an infinite number of rectangles.)

In integration, 'dx' represents the width of the rectangles which you are summing. And since there are an 'infinite number' of these rectangles, dx is infinitely small.


Another way to think about the concept of differentials is to think back to the formula for slope.

\(\displaystyle slope = \frac{{y_2 - y_1 }}{{x_2 - x_1 }} = \frac{{\Delta y}}{{\Delta x}}\)

Find any two points of a function, you can find the slope of the secant. However, if the change in x becomes very very small, the slope of the secant approaches the slope of the tangent.

Compare the slope formula with the first principals definition for a derivative:

\(\displaystyle {\lim }\limits_{h - > 0} \frac{{f(x + h) - f(x)}}{h}\)

Differentiating is the process of finding the derivative of a given function. The derivative is a formula that gives the slope at any point on the original function.

 

[xailer]

Well your replies are a bit too advanced for me since all I know about integrals is what I told in my first post (and I really do mean "all I know" ).Can you dumb it down a little?

First I want to warn you not to conflate the idea of ‘Integral’ and ‘antiderivative’.
While they are related they are most definitely not the same.
In higher mathematics an integral is always a number.

What is the difference between anti-derivative and indefinite integral?



I'm gonna quote my text book ( actually I will try to quote it since I have to translate to english ): "From examples we saw that initial function with its derivative i.e. differential is not precisely fixed. With every constant C:

(F(x)+C)'= F'(x)+C = f(x)

i.e.

d(F(x)+C) = dF(x) = f(x)dx

"

NOTE: I'm not actually sure that word "i.e" I used two times in a quote is correct translation (instead I could use word "actually" or "more precisely" or something to that effect ). Actually I'm not preciselly sure myself how to interpret (in my own language ) that word in that context ( perhaps you will be able to figure it out ).


So what is going on? First formula "(F(x)+C)'= F'(x)+C = f(x)" makes prefect sense but second d(F(x)+C) = dF(x) = f(x)dx seems somehow out of place. Can you explain (in simple terms) the meaning of second formula and how it relates to first formula?




"We compute derivative F' (differential of function dF) for function F with derivating
( differentiating ) "

NOTE: I'm not sure there is an english word for finding derivative - so I call it derivating

Anyhow, why is there alternative word in parenthesis [ derivative F'-->(differential of function dF) and derivating-->( differentiating ) ], as if the two words mean the same (which they don't)?

In fact I studied integration theory with someone who made all his students write \(\displaystyle \int {f(x)}\) without the dx.

Is the point you're trying to make that I should put dx in formula \(\displaystyle \int {f(x)} dx\), but ignore the reasons dx was put there and pretend it isn't there (at least until I get to more advanced math)?

Another way to think about the concept of differentials is to think back to the formula for slope.

\(\displaystyle slope = \frac{{y_2 - y_1 }}{{x_2 - x_1 }} = \frac{{\Delta y}}{{\Delta x}}\)

Find any two points of a function, you can find the slope of the secant. However, if the change in x becomes very very small, the slope of the secant approaches the slope of the tangent.

Compare the slope formula with the first principals definition for a derivative:

\(\displaystyle {\lim }\limits_{h - > 0} \frac{{f(x + h) - f(x)}}{h}\)

Differentiating is the process of finding the derivative of a given function. The derivative is a formula that gives the slope at any point on the original function.

I understand the concept of differentials in that context.





This is called differentianting:

F(x) --> dF(x) = f(x)dx

and this integrating:

f(x)dx --> integral[ f(x)dx ] = F(x) ?


I simply can not see the correlation between my definition of integrals in my first post and differentials!

 

[pka]

“What is the difference between anti-derivative and indefinite integral?”
Put that way, there is no difference.
But my graduate advisor would say, “There is not such a thing as an indefinite integral!” An integral is a number. Again that is graduate mathematics.

I suppose that every calculus textbook uses that term and we all know what is intended when it is used: find the anti-derivative.

“the point you're trying to make that I should put dx in formula” is to tell us the variable of integration.

“This is called differentianting:
F(x) --> dF(x) = f(x)dx YES!

and this integrating:
f(x)dx --> integral[ f(x)dx ] = F(x) ? NO!
In Cal I&II it is call an indefinite integral.
In vector calculus, it is called ‘finding the primitive’ .

 

[Razor X]

I’m sure all of those replies are perfectly valid, however advanced they are. But, I had this same problem: why does this dx symbol keep appearing and what does it really mean. Finally, I found something that put it in historical context.

Basically I found out that the notation dx is a leftover from infinitesimal calculus. Modern calculus is presented using limits as its foundation, but long ago before limits were understood, calculus was based on infinitesimals. The notation dx is an infinitesimal number.

But, since limits are now accepted as a better foundation, any meaning dx had is obsolete. That’s why dy/dx is the symbol for the derivative, but we can’t think of it as dy divided by dx any more. Also, the same idea holds for integral notation, but dx can’t be thought of being multiplied (except in the case of u substitution).

As far as I know, my calculus book still defines a differential as, if y=f(x) then dy=f’(x)dx but they use a geometric definition to give meaning to the symbols.

Anyway, that’s just my quick summary of what’s going on, but if you want more info, I suggest searching Google or wikipedia for infinitesimal calculus.

Oh, and also, the dx in the integral is related to the (delta)x in a Riemann sum, but only in a conceptual way. So its main purpose is to identify the variable, just like dx identifies the variable in the derivative.