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I have a math book full of exercises and it doesn't explain at all how a integral works.It just shows me some integrals that I learned in highschool and most of them don't even show the proof behind them.

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I have a math book full of exercises and it doesn't explain at all how a integral works.It just shows me some integrals that I learned in highschool and most of them don't even show the proof behind them.

- #2

mfb

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If f'(x) is the derivative of f(x), then f(x) is an integral function of f'(x).

You already encountered cases of this much earlier: If 5+6=11, then 11-6=5. Subtracting 6 is the inverse action of adding 6. If 3*4=12, then 12/4=3. Dividing by 4 is the inverse of multiplying by 4.

If you add a constant to a function, its derivative does not change: f(x) and f(x)+3 have the same derivative. If you are only given the derivative, you don't know if it is the derivative of f(x), of f(x)+6423, or something else. Therefore the +c is added to keep all options. Every value of c leads to a possible integral function.

- #3

Stephen Tashi

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Begin by making a distinction between "antiderivative" and "definite integral". Some people use the term "integral" to refer to both those concepts, but the two concepts have different definitions. Informally, a "definite integral" of a function refers to an area "under" its graph. The antiderivative of a function f(x) is another function F(x) such that F'(x) = f(x). For example, if f(x) = 2x, then both F_a(x) = x^2 and F_b(x) = x^2 + 6 are antiderivatives of f(x).when it comes to integrals and their uses I do not understand what they do and where you use them.

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The connection between definite integrals and antiderivatives is given by The Fundmental Theorem of Calculus, which you should know if you have taken a calculus course. It's normal for a student not to have an intuitive understanding of why The Fundamental Theorem of Calculus should be true, but there's no excuse for not knowing what that theorem says and how it connects the idea of definite integral with the idea of antiderivative. Do you understand what the Fundamental Theorem of Calculus says?

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Stephen Tashi

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You probably mean "...are the antiderivatives...".the integral of some functions are the derivatives of that function is not.

You didn't say whether you know what the Fundamental Theorem Of Calculus says.

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- #7

fresh_42

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If you dig to the bottom of this question, the answer will be: because multiplication is harder than addition.

To build a derivative, we attach a ruler at some point of the object, which is a linear, i.e.an additive thing. To integrate, we have to calculate a volume, which is a multiplication, or many small ones in this case. The formula ##e^x \cdot e^y = e^{x+y}## illustrates the difficulty: the derivative corresponds to the addition in the exponent, while the integration is the multiplication on the base line. That's why most differential equation are solved by searching for something like ##c(x)e^{a(x)}##.

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Mark44

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Differentiation for many functions is very straightforward -- you use the product rule, chain rule, etc., and you might have to use several rules in succession in a certain order. Going backward (antidifferentiating) is much less straightforward, since the procedure is much less "cookbook" than differentiation.

Whether you know it or not, integration by parts is the reverse operation of the product rule. If h(x) = f(x) * g(x), then h'(x) = f(x) * g'(x) + f'(x) g(x). This equation can be transformed to f(x) * g'(x) = h'(x) - f'(x) * g(x). If you multiply both sides by dx and antidifferentiate both sides you get ##\int f(x) g'(x) dx = h(x) - \int f'(x) g(x) dx##, which is integration by parts.

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Mark44

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In English we say you canThe truth is that you can derivate functions

Derivate is a word, but not one that's used in discussions of calculus.

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Stephen Tashi

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Have you taken a calculus course? You seem unfamiliar with standard terminology. We "differentiate" functions , not "derivate" them.The truth is that you can derivate functions but not all functions are the derivative of a function.

If you understood the concepts of integral and derivative, you'd realize you haven't given a specific example.For ex the integral of e^x/x has no derivative.

I have to guess at your level of understanding. The fact that you are carless with terminology - such as writing "derivative" when you mean "antiderivative" suggests that your are approaching calculus as set of procedures for manipulating symbols - i.e. as a more sophisticated version of manipulations that are done in elementary algebra. You appear frustrated because the manipulations permitted for antidifferentiation are more complicated that those permitted for differentiation. You won't fully understand why certain procedures of manipulating symbols are pemitted unless you understand the verbal concepts that underlie the manipulations. If you are using a standard calculus text, there are verbal explanations for the concepts of definite integral and antiderivative. So my guess is that you are ignoring the explanations in your text materials. People on this forum can present their own versions of what's aleady in your textbooks and perhaps the personal attention will motivate you to pay closer attention to these ideas.Now I'm really confused.I just don't see the logic of integrals in derivatives.Can you explain me that?

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fresh_42

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I struggle between derivative and derivation. It's the same at its core, but then again different in context.In English we say you candifferentiatefunctions.

Derivate is a word, but not one that's used in discussions of calculus.

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Sorry I meant differentiate.

- #14

Mark44

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I'm sure it's confusing, but then English is not known for being logical.I struggle between derivative and derivation. It's the same at its core, but then again different in context.

As far as derivative and derivation, it's just something you have to remember, since the usage isn't logical at all.

The word

You can start with a quadratic equation, and use completing the square to

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Mark44

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No problem and no need for an apology. I wasn't trying to be critical. I know you're not a native speaker of English. I was just trying to help you out.Sorry I meant differentiate.

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Uh no one knows?

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fresh_42

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No that isn't it. I've studied a lot about Lie algebras, and they have derivations, which are linear mappings defined by ##d([a,b])=[d(a),b]+[a,d(b)]##. This is just a version of the Leibniz rule, as the Jacobi identity is, because it is deduced - or should I say derived - from derivatives. It's basically the same thing, just not calculus. And as I'm more used to write derivation, it occasionally slips into derivatives. It would be equally difficult in German, but the German word for derivative is "Ableitung". It's the literal translation of the Latin one, so it's a bit easier because of that.I'm sure it's confusing, but then English is not known for being logical.

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fresh_42

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I do. Post number seven.Uh no one knows?

- #21

Mark44

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Sure, these kinds of definite integrals come up fairly often. For example, ##\int_0^\infty e^{-x}dx## is a fairly simple example of a convergent improper integral whose value is 1.

In contrast, the improper integral ##\int_{-\infty}^0 e^{-x}dx## diverges.

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fresh_42

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I don't quite understand. Integrals

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- #25

FactChecker

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Some real-world examples may be helpful. The integral of a function is the accumulation of the function values over a range of its inputs. If you know the velocity of a car as a function of time, v(t), then you can integrate it to keep track of the position of a car whose initial position at time 0 is ##p_0##, ##p(t) = p_0 + \int_0^t v(s)ds##. (You would need to do that for every dimension that the car can travel in.) One step further, you can accumulate the accelerations of a car to keep track of its velocity. This is a very common application because accelerometers are very easy to include in a vehicle.

A function with a derivative at a point is very well behaved at that point and around it. The derivative is only a local property and the behavior of the function some distance from that point are irrelevant. The function will be continuous at the point where it has a derivative.

Very bazarre functions have integrals. The functions may be extremely discontinuous. Furthermore, since the integral is an accumulation of the function values over a range of input values, it depends on the behavior of the function over the entire range. That is a much more difficult thing to handle than the local property of a derivative.

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