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I've realised recently that despite knowing how to integrate different kind of functions, I've never really learned exactly what I'm doing precisely when I'm integrating. Obviously, I'm finding the signed area under the curve of a function (or between bound if the integral is definite), but I couldn't explain why $$\int \frac{1}{x} dx$$ Is equal to $ln |x| + c$ other than stating that the derivative of $ln\ x$ is $\frac {1}{x}$ and so it's the antiderivative. The definition of the derivative is quite concise and elegant, using the limit definition of the derivative, which can explain itself in the function itself, but I've never heard of such a thing for integration.

Integration is a unique mathematical operation for me where there's no intuition at all in its solutions. The fact that

$$\int \frac{1}{\sqrt{1-x^2}}dx$$

is a much more straightforward computation than

$$\int \sqrt{x^2 + 1} \ dx$$

is something I find a bit unintuitive. Another example is the integral of $\int sec\ x \ dx$ for example. It's astonishingly complex for such a base function.

Plus, combining the two explanations I have for integration can produce some odd results. Integration is the signed area under a curve, but it's also the antiderivative of a function. For instance, the antiderivative of $cos \ x$ is $sin \ x + c$, but why on earth is $sin\ x$ a function that is going to help us evaluate the area under the curve of a cosine function? Where's the intuition in that?

Essentially, other than it being the reverse process of certain ways to find derivatives, hence it being an antiderivative, that can be explained with what I find an elegant and intuitive formula, is there such a thing for integration as well to explain its interesting characteristics?

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It's the FTC stupid! (This was only a joke).

Denote the "area under the curve" by $$\int_{[a,b]} f(t)\>{\rm d}t$$ for the moment. This area has an intuitive geometric description and can be mathematically defined as limit of Riemann sums: $$\int_{[a,b]} f(t)\>{\rm d}t:=\lim_{\ldots}\sum_{k=1}^Nf(\tau_k)(t_k-t_{k-1})\ .\tag{1}$$ Only for very special functions, e.g., $x\mapsto e^{\lambda x}$, we can compute such a limit directly. Now there comes a radically new idea: Consider the function $$F_a(x):=\int_{[a,x]} f(t)\>{\rm d}t\qquad(a\leq x\leq b)\ ,$$ where now the upper end point of the interval of integration is variable. Using the consequences of the definition $(1)$ one proves that $$F_a'(x)={d\over dx}\int_{[a,x]} f(t)\>{\rm d}t=f(x)\qquad (a\leq x\leq b)\ .\tag{2}$$ This is the "first part" of the FTC. Going on one then shows the "second part": If $F$ is any antiderivative of $f$ on the interval $[a,b]$ then $$\int_{[a,b]} f(t)\>{\rm d}t=\int_a^b f(t)\>dt:=F(b)-F(a)\ .$$ The first equality sign here is not a tautology in disguise, but the very essence of the FTC: The limit of Riemann sums is equal to the total increment of the antiderivatives between $a$ and $b$.

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  • $\begingroup$ (+1), but for posterity it might be worth explicitly assuming $f$ is continuous, at least in (2)...? $\endgroup$ – Andrew D. Hwang Jul 1 '17 at 10:15
  • $\begingroup$ No offence given, thanks for your answer! I'll need to take a look at Riemann sums, I reckon. $\endgroup$ – sangstar Jul 1 '17 at 15:17
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I would like to complement the other excellent answer mentioning the FTC with a detail to compare the definitions of derivative and integral.

The OP claims the definition of the derivative by a limit is concise and elegant. But pause here for a moment and think there is much more. The fact is that we seldom use the definition, but the formulas we learn by heart for $f'(x)$. So there is a hidden process of abstraction we usually skim:

  1. First we define a number that is the derivative $M_a$ at a point $a$ by the usual limit definition. Geometrically, this amounts to drawing the tangent to the curve, which is a straight line.
  2. Then we construct a function $f'$ that assigns to each point $x$ the derivative at that point $M_x$. Only now are we allowed to write $f'(a)=\lim \dots$, there was no such function $f'(x)$ before. Geometrically this function $f'(x)$ can be complicated and, in general, is not a straight line.

Mathematically these two objects are completely different. Now pass on to the integral, which is further complicated by using the same notation $\int$ for two different objects:

  1. First you define a number that measures the area under a curve between points $a$ and $b$. As with the derivative, this definition uses a limit (of Riemann sums).
  2. Then you construct a function that assigns to each point $x$ the area between $a$ and $x$.

Up to now there is no theoretical result, just definitions. But now the FTC is easy to state: the derivative of the function in 2. is the function of the curve in 1. Observe that I intentionally avoid the definition of antiderivative, indefinite integral or primitive.

And a final point: the notion of integrability has nothing to do with the computation of primitives in terms of elementary functions. In other words, I do not know either why there are "easy" and "difficult" (or impossible) primitives.

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