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Today is the advanced mathematics topics first eight articles, today's content is an indefinite integral.
I had a high number of teacher once said that mathematics is more than half of this with some knowledge of calculus and limits the number of columns. The calculus of them, relevant and integral to occupy most of the country. Calculus is important not because it's heavy, more capacity, but because it is commonly used . Almost textbook on calculus formula has all science and engineering, the reason is very simple, when the scientists in the study of the unknown or the calculated time, extensive use of the calculus as a tool. This is also the reason we have to learn it.
Primitive
I have always felt the calculus of this name was very good, calculus differentiation and integration are collectively known . Differential by macro-microscopic study, while integrating the contrary is acquired by the macro-micro. So in a sense, we can be seen as negative points differential.
Differential corresponds to the limit, the function of which, let us study the change of function in 0 approach. As
a function of the rate of change tends to zero, we get is the derivative of the function, which is the derivative formula origin:
We integral differential from the point of view, that we think the reverse of this process. If the derivative we get that , then the previous derivation function f (x) what would it be? In this issue among the function before the derivation called the original function , we write F (x), if F (x) is f (x) of the original function, then it should meet for any
, there is
.
For example, because the derivative is 2x, so
it is 2x the original function.
Relations function and the original function we know, but in order to rigorous, we also need to think about a problem, the original function must exist yet ?
This issue looks around, in fact, very easy figured, if the function is continuous, then the original function must exist . It says the high number of this book is the original function existence theorem, but did not prove that even a word, we can imagine it basically has been treated as an axiom. Let's look at a simple analysis, such as the function f (x) continuous, that is the derivative of the original function of the present and continuous. We know that can not necessarily consecutive guide, but the guide must be continuous. And now there is a continuous function guide, then the description of the original function must be continuous. If the function does not exist how continuous it? Therefore, the current function f (x) continuously, indicating that the primitive F (x) must exist.
indefinite integral
We figured it out after the original function, you can start the indefinite integral of the content. In fact, nothing indefinite integral calculation of content, I would think more like a map. The current function mapped to the original function.
In other words, we have to find a primitive function F (x) by the current function f (x), such that: We call this process is reversed to write, namely:
This equation is actually a derivation of the inverse operation , there is no technical content, should all slow connection. This time, we ask a question, for a certain function f (x), for its original function is to determine it?
For example, we have just that example , its original function only
do?
The answer is obvious, it is not . We just would include another original function: Similarly, we put back into other constant value is the same as the original legitimate function. So we can know, the original function is infinite, the only difference is last with different constants. That original function because there is this constant uncertainty, which is among the indefinite integral " uncertain origin of" the word.
Simple Properties
According to the definition of indefinite integral, we can derive some simple nature. Let's first look at nature, nature is also the simplest:
This proved to be very simple, we direct derivation of the original formula to:
There is another equally simple nature:
Proof and just as direct derivation can be.
Well, that's the nature of the indefinite integral all. You may ask why there is no nature nature multiplication and division? I have also curious about this issue, because all the information I checked nice of them are not related to the formula. I have also tried to deduce too, but no results. This is certainly not a mathematician who is lazy or can not be calculated, estimates may be too complicated, so it is not practical.
Basic Standings
Finally, we look at the basic standings indefinite integral, easy when we compute queries.
$$ \begin{aligned} \int kdx &= kx+C \\ \int x^{\mu}dx &= \frac{x^{\mu+1}}{\mu+1}+C \\ \int \frac{dx}{x} &= \ln|x| + C \\ \int \frac{dx}{1+x^2} &= \arctan x + C\\ \int \frac{dx}{1-x^2} &= \arcsin x + C\\ \int \cos x dx &= \sin x + C\\ \int \sin x dx &= -\cos x + C\\ \int \frac{dx}{\cos^2x}&=\int \sec^2x dx = \tan x + C \\ \int \frac{dx}{\sin^2 x} &= \int \csc^2 x dx = -\cot x + C\\ \int \sec x \tan x dx &= \sec x + C\\ \int \csc x \cot x dx &= -\csc x + C\\ \int e^xdx &= e^x + C\\ \int a^x dx &= \frac{a^x}{\ln a} + C \end{aligned} $$不定积分本身的内容就是这么多，理解起来并不困难。不过在实际解决问题的过程当中，还存在一些解题的技巧，由于篇幅问题，我们放到下一篇文章当中和大家一起分享。
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