Integrals for dummies: how to solve, calculation rules, explanation. A function F(x) is called an antiderivative of a function f(x) if F`(x)=f(x) or dF(x)=f(x)dx
Some interval X. If For any xХ F"(x) = f(x), then function F calledantiderivativeForfunctions f on the interval X. AntiderivativeForfunctions you can try to find...
Antiderivative for function
Document... . Function F(x) calledantiderivativeForfunctions f(x) on the interval (a;b), if For all x(a;b) the equality F(x) = f(x) holds. For example, Forfunctions x2 antiderivative will function x3...
Fundamentals of Integral Calculus Study Guide
Tutorial... ; 5. Find the integral. ; B) ; C) ; D) ; 6. Functioncalledantiderivative To functions on a set if: For everyone; at some point; For everyone; at some... interval. Definition 1. FunctioncalledantiderivativeForfunctions on many...
Antiderivative Indefinite integral
DocumentIntegration. Antiderivative. Continuous function F(x) calledantiderivativeForfunctions f (x) on the interval X if For each F’ (x) = f (x). EXAMPLE Function F(x) = x 3 is antiderivativeForfunctions f(x) = 3x...
SPECIAL EDUCATION OF THE USSR Approved by the Educational and Methodological Directorate for Higher Education HIGHER MATHEMATICS METHODICAL INSTRUCTIONS AND CONTROL TASKS (WITH THE PROGRAM) for part-time students of engineering and technical specialties
GuidelinesQuestions For self-test Define antiderivativefunctions. Indicate the geometric meaning of the aggregate primitivefunctions. What called uncertain...
We have seen that the derivative has numerous uses: the derivative is the speed of movement (or, more generally, the speed of any process); derivative is the slope of the tangent to the graph of the function; using the derivative, you can examine a function for monotonicity and extrema; the derivative helps solve optimization problems.
But in real life Inverse problems also have to be solved: for example, along with the problem of finding the speed according to a known law of motion, there is also the problem of restoring the law of motion according to a known speed. Let's consider one of these problems.
Example 1. A material point moves in a straight line, its speed at time t is given by the formula u = tg. Find the law of motion.
Solution. Let s = s(t) be the desired law of motion. It is known that s"(t) = u"(t). This means that to solve the problem you need to choose function s = s(t), whose derivative is equal to tg. It's not hard to guess that
Let us immediately note that the example is solved correctly, but incompletely. We found that, in fact, the problem has infinitely many solutions: any function of the form 
an arbitrary constant can serve as a law of motion, since
To make the task more specific, we needed to fix the initial situation: indicate the coordinate of a moving point at some point in time, for example, at t=0. If, say, s(0) = s 0, then from the equality we obtain s(0) = 0 + C, i.e. S 0 = C. Now the law of motion is uniquely defined:
In mathematics, mutually inverse operations are given different names and special notations are invented: for example, squaring (x 2) and extracting square root sine(sinх) and arcsine(arcsin x), etc. The process of finding the derivative of a given function is called differentiation, and the inverse operation, i.e. the process of finding a function from a given derivative - integration.
The term “derivative” itself can be justified “in everyday life”: the function y - f(x) “gives birth” to a new function y"= f"(x). The function y = f(x) acts as a “parent” , but mathematicians, naturally, do not call it a “parent” or “producer”; they say that this, in relation to the function y"=f"(x), is the primary image, or, in short, the antiderivative.
Definition 1. The function y = F(x) is called antiderivative for the function y = f(x) on a given interval X if for all x from X the equality F"(x)=f(x) holds.
In practice, the interval X is usually not specified, but is implied (as the natural domain of definition of the function).
Here are some examples:
1) The function y = x 2 is antiderivative for the function y = 2x, since for all x the equality (x 2)" = 2x is true.
2) the function y - x 3 is antiderivative for the function y-3x 2, since for all x the equality (x 3)" = 3x 2 is true.
3) The function y-sinх is antiderivative for the function y = cosx, since for all x the equality (sinx)" = cosx is true.
4) The function is antiderivative for a function on the interval since for all x > 0 the equality is true
In general, knowing the formulas for finding derivatives, it is not difficult to compile a table of formulas for finding antiderivatives.
We hope you understand how this table is compiled: the derivative of the function, which is written in the second column, is equal to the function that is written in the corresponding row of the first column (check it, don’t be lazy, it’s very useful). For example, for the function y = x 5 the antiderivative, as you will establish, is the function (see the fourth row of the table).
Notes: 1. Below we will prove the theorem that if y = F(x) is an antiderivative for the function y = f(x), then the function y = f(x) has infinitely many antiderivatives and they all have the form y = F(x ) + C. Therefore, it would be more correct to add the term C everywhere in the second column of the table, where C is an arbitrary real number.
2. For the sake of brevity, sometimes instead of the phrase “the function y = F(x) is an antiderivative of the function y = f(x),” they say F(x) is an antiderivative of f(x).”
2. Rules for finding antiderivatives
When finding antiderivatives, as well as when finding derivatives, not only formulas are used (they are listed in the table on p. 196), but also some rules. They are directly related to the corresponding rules for calculating derivatives.
We know that the derivative of a sum is equal to the sum of its derivatives. This rule generates the corresponding rule for finding antiderivatives.
Rule 1. The antiderivative of a sum is equal to the sum of the antiderivatives.
We draw your attention to the somewhat “lightness” of this formulation. In fact, one should formulate the theorem: if the functions y = f(x) and y = g(x) have antiderivatives on the interval X, respectively y-F(x) and y-G(x), then the sum of the functions y = f(x)+g(x) has an antiderivative on the interval X, and this antiderivative is the function y = F(x)+G(x). But usually, when formulating rules (and not theorems), they leave only keywords- this makes it more convenient to apply the rule in practice
Example 2. Find the antiderivative for the function y = 2x + cos x.
Solution. The antiderivative for 2x is x"; the antiderivative for cox is sin x. This means that the antiderivative for the function y = 2x + cos x will be the function y = x 2 + sin x (and in general any function of the form Y = x 1 + sinx + C) .
We know that the constant factor can be taken out of the sign of the derivative. This rule generates the corresponding rule for finding antiderivatives.
Rule 2. The constant factor can be taken out of the sign of the antiderivative.
Example 3.
Solution. a) The antiderivative for sin x is -soz x; This means that for the function y = 5 sin x the antiderivative function will be the function y = -5 cos x.
b) The antiderivative for cos x is sin x; This means that the antiderivative of a function is the function
c) The antiderivative for x 3 is the antiderivative for x, the antiderivative for the function y = 1 is the function y = x. Using the first and second rules for finding antiderivatives, we find that the antiderivative for the function y = 12x 3 + 8x-1 is the function
Comment. As is known, the derivative of a product is not equal to the product of derivatives (the rule for differentiating a product is more complex) and the derivative of a quotient is not equal to the quotient of derivatives. Therefore, there are no rules for finding the antiderivative of the product or the antiderivative of the quotient of two functions. Be careful!
Let us obtain another rule for finding antiderivatives. We know that the derivative of the function y = f(kx+m) is calculated by the formula
This rule generates the corresponding rule for finding antiderivatives.
Rule 3. If y = F(x) is an antiderivative for the function y = f(x), then the antiderivative for the function y=f(kx+m) is the function
Indeed,
This means that it is an antiderivative for the function y = f(kx+m).
The meaning of the third rule is as follows. If you know that the antiderivative of the function y = f(x) is the function y = F(x), and you need to find the antiderivative of the function y = f(kx+m), then proceed like this: take the same function F, but instead of the argument x, substitute the expression kx+m; in addition, do not forget to write “correction factor” before the function sign
Example 4. Find antiderivatives for given functions:
Solution, a) The antiderivative for sin x is -soz x; This means that for the function y = sin2x the antiderivative will be the function
b) The antiderivative for cos x is sin x; This means that the antiderivative of a function is the function
c) The antiderivative for x 7 means that for the function y = (4-5x) 7 the antiderivative will be the function
3. Indefinite integral
We have already noted above that the problem of finding an antiderivative for a given function y = f(x) has more than one solution. Let's discuss this issue in more detail.
Proof. 1. Let y = F(x) be the antiderivative for the function y = f(x) on the interval X. This means that for all x from X the equality x"(x) = f(x) holds. Let us find the derivative of any function of the form y = F(x)+C:
(F(x) +C) = F"(x) +C = f(x) +0 = f(x).
So, (F(x)+C) = f(x). This means that y = F(x) + C is an antiderivative for the function y = f(x).
Thus, we have proven that if the function y = f(x) has an antiderivative y=F(x), then the function (f = f(x) has infinitely many antiderivatives, for example, any function of the form y = F(x) +C is an antiderivative.
2. Let us now prove that the indicated type of functions exhausts the entire set of antiderivatives.
Let y=F 1 (x) and y=F(x) be two antiderivatives for the function Y = f(x) on the interval X. This means that for all x from the interval X the following relations hold: F^ (x) = f (X); F"(x) = f(x).
Let's consider the function y = F 1 (x) -.F(x) and find its derivative: (F, (x) -F(x))" = F[(x)-F(x) = f(x) - f(x) = 0.
It is known that if the derivative of a function on an interval X is identically equal to zero, then the function is constant on the interval X (see Theorem 3 from § 35). This means that F 1 (x) - F (x) = C, i.e. Fx) = F(x)+C.
The theorem has been proven.
Example 5. The law of change of speed with time is given: v = -5sin2t. Find the law of motion s = s(t), if it is known that at time t=0 the coordinate of the point was equal to the number 1.5 (i.e. s(t) = 1.5).
Solution. Since speed is a derivative of the coordinate as a function of time, we first need to find the antiderivative of the speed, i.e. antiderivative for the function v = -5sin2t. One of such antiderivatives is the function , and the set of all antiderivatives has the form:
To find the specific value of the constant C, we use the initial conditions, according to which s(0) = 1.5. Substituting the values t=0, S = 1.5 into formula (1), we get:
Substituting the found value of C into formula (1), we obtain the law of motion that interests us:
Definition 2. If a function y = f(x) has an antiderivative y = F(x) on an interval X, then the set of all antiderivatives, i.e. the set of functions of the form y = F(x) + C is called the indefinite integral of the function y = f(x) and is denoted by:
(read: “indefinite integral ef from x de x”).
In the next paragraph we will find out what the hidden meaning of this designation is.
Based on the table of antiderivatives available in this section, we will compile a table of the main indefinite integrals:
Based on the above three rules for finding antiderivatives, we can formulate the corresponding integration rules.
Rule 1. Integral of the sum of functions equal to the sum integrals of these functions:
Rule 2. The constant factor can be taken out of the integral sign:
Rule 3. If
Example 6. Find indefinite integrals:
Solution, a) Using the first and second rules of integration, we obtain:
Now let's use the 3rd and 4th integration formulas:
As a result we get:
b) Using the third rule of integration and formula 8, we obtain:
c) To directly find a given integral, we have neither the corresponding formula nor the corresponding rule. In such cases, pre-executed identity transformations expression contained under the integral sign.
Let's use the trigonometric formula for reducing the degree:
Then we find sequentially:
A.G. Mordkovich Algebra 10th grade
Calendar-thematic planning in mathematics, video in mathematics online, Mathematics at school
Indefinite integral
The main task of differential calculus was to calculate the derivative or differential of a given function. Integral calculus, to the study of which we are moving on, solves the inverse problem, namely, finding the function itself from its derivative or differential. That is, having dF(x)= f(x)d (7.1) or F ′(x)= f(x),
Where f(x)- known function, need to find the function F(x).
Definition:The function F(x) is called antiderivative function f(x) on the segment if the equality holds at all points of this segment: F′(x) = f(x) or dF(x)= f(x)d.
For example, one of the antiderivative functions for the function f(x)=3x 2 will F(x)= x 3, because ( x 3)′=3x 2. But a prototype for the function f(x)=3x 2 there will also be functions and , since 
.
So, this function f(x)=3x 2 has an infinite number of primitives, each of which differs only by a constant term. Let us show that this result also holds in the general case.
Theorem Two different antiderivatives of the same function defined in a certain interval differ from each other on this interval by a constant term.
Proof
Let the function f(x) defined on the interval (a¸b) And F 1 (x) And F 2 (x) - antiderivatives, i.e. F 1 ′(x)= f(x) and F 2 ′(x)= f(x).
Then F 1 ′(x)=F 2 ′(x)Þ F 1 ′(x) - F 2 ′(x) = (F 1 ′(x) - F 2 (x))′= 0. Þ F 1 (x) - F 2 (x) = C
From here, F 2 (x) = F 1 (x) + C
Where WITH - constant (a corollary of Lagrange’s theorem is used here).
The theorem is thus proven.
Geometric illustration. If at = F 1 (x) And at = F 2 (x) – antiderivatives of the same function f(x), then the tangent to their graphs at points with a common abscissa X parallel to each other (Fig. 7.1).
In this case, the distance between these curves along the axis OU remains constant F 2 (x) - F 1 (x) = C , that is, these curves in some understanding"parallel" to one another.
Consequence .
Adding to some antiderivative F(x) for this function f(x), defined on the interval X, all possible constants WITH, we get all possible antiderivatives for the function f(x).So the expression F(x)+C , where , and F(x) – some antiderivative of a function f(x) includes all possible antiderivatives for f(x).
Example 1. Check if functions are antiderivatives of the function
Solution:
Answer: antiderivatives for a function there will be functions And
Definition: If the function F(x) is some antiderivative of the function f(x), then the set of all antiderivatives F(x)+ C is called indefinite integral of f(x) and denote:
∫f(х)dх.
A-priory:
f(x) - integrand function,
f(х)dх - integrand expression
It follows from this that the indefinite integral is a function of general form, the differential of which is equal to the integrand, and the derivative of which with respect to the variable X is equal to the integrand at all points.
From a geometric point of view an indefinite integral is a family of curves, each of which is obtained by shifting one of the curves parallel to itself up or down, that is, along the axis OU(Fig. 7.2).
The operation of calculating the indefinite integral of a certain function is called integration this function.
Note that if the derivative of an elementary function is always an elementary function, then the antiderivative of an elementary function may not be represented by a finite number of elementary functions.
Let's now consider properties of the indefinite integral.
From Definition 2 it follows:
1. The derivative of the indefinite integral is equal to the integrand, that is, if F′(x) = f(x) , That
2. The differential of the indefinite integral is equal to the integrand
. (7.4)
From the definition of differential and property (7.3)
3. The indefinite integral of the differential of some function is equal to this function up to a constant term, that is 
(7.5)
There are three basic rules for finding antiderivative functions. They are very similar to the corresponding differentiation rules.
Rule 1
If F is an antiderivative for some function f, and G is an antiderivative for some function g, then F + G will be an antiderivative for f + g.
By definition of an antiderivative, F’ = f. G' = g. And since these conditions are met, then according to the rule for calculating the derivative for the sum of functions we will have:
(F + G)’ = F’ + G’ = f + g.
Rule 2
If F is an antiderivative for some function f, and k is some constant. Then k*F is the antiderivative of the function k*f. This rule follows from the rule for calculating the derivative complex function.
We have: (k*F)’ = k*F’ = k*f.
Rule 3
If F(x) is some antiderivative for the function f(x), and k and b are some constants, and k is not equal to zero, then (1/k)*F*(k*x+b) will be an antiderivative for the function f (k*x+b).
This rule follows from the rule for calculating the derivative of a complex function:
((1/k)*F*(k*x+b))’ = (1/k)*F’(k*x+b)*k = f(k*x+b).
Let's look at a few examples of how these rules apply:
Example 1. Find general form antiderivatives for the function f(x) = x^3 +1/x^2. For the function x^3 one of the antiderivatives will be the function (x^4)/4, and for the function 1/x^2 one of the antiderivatives will be the function -1/x. Using the first rule, we have:
F(x) = x^4/4 - 1/x +C.
Example 2. Let's find the general form of antiderivatives for the function f(x) = 5*cos(x). For the function cos(x), one of the antiderivatives will be the function sin(x). If we now use the second rule, we will have:
F(x) = 5*sin(x).
Example 3. Find one of the antiderivatives for the function y = sin(3*x-2). For the function sin(x) one of the antiderivatives will be the function -cos(x). If we now use the third rule, we obtain an expression for the antiderivative:
F(x) = (-1/3)*cos(3*x-2)
Example 4. Find the antiderivative for the function f(x) = 1/(7-3*x)^5
The antiderivative for the function 1/x^5 will be the function (-1/(4*x^4)). Now, using the third rule, we get.
Prototype. Beautiful word.) First, a little Russian. This word is pronounced exactly like this, not "prototype" , as it may seem. Antiderivative is the basic concept of all integral calculus. Any integrals - indefinite, definite (you will become familiar with them this semester), as well as double, triple, curvilinear, surface (and these are already the main characters of the second year) - are built on this key concept. Makes complete sense to master. Go.)
Before getting acquainted with the concept of an antiderivative, let us first general outline let's remember the most common one derivative. Without delving into the boring theory of limits, argument increments and other things, we can say that finding the derivative (or differentiation) is simply a mathematical operation on function. That's all. Any function is taken (for example, f(x) = x2) And By certain rules transforms into new feature. And this one is the one new feature and is called derivative.
In our case, before differentiation there was a function f(x) = x2, and after differentiation it became already other function f’(x) = 2x.
Derivative– because our new function f’(x) = 2x happened from function f(x) = x2. As a result of the differentiation operation. And specifically from it, and not from some other function ( x 3, For example).
Roughly speaking, f(x) = x2- this is mom, and f’(x) = 2x- her beloved daughter.) This is understandable. Go ahead.
Mathematicians are restless people. For every action they try to find a reaction. :) There is addition - there is also subtraction. There is multiplication and there is division. Raising to a power is extracting the root. Sine - arcsine. Exactly the same differentiation- that means there is... integration.)
Now let's put this interesting task. For example, we have such a simple function f(x) = 1. And we need to answer this question:
The derivative of WHAT function gives us the functionf(x) = 1?
In other words, seeing a daughter, using DNA analysis, figure out who her mother is. :) So from which one? original function (let's call it F(x)) our derivative function f(x) = 1? Or in mathematical form, for which function F(x) the following equality holds:
F’(x) = f(x) = 1?
An elementary example. I tried.) We simply select the function F(x) so that the equality works. :) Well, did you find it? Yes, sure! F(x) = x. Because:
F’(x) = x’ = 1 = f(x).
Of course, the found mommy F(x) = x I need to call it something, yes.) Meet me!
Antiderivative for functionf(x) such a function is calledF(x), whose derivative is equal tof(x), i.e. for which the equality holdsF’(x) = f(x).
That's all. No more scientific tricks. In the strict definition, an additional phrase is added "on the interval X". But we will not delve into these subtleties for now, because our primary task is to learn to find these very primitives.
In our case, it turns out that the function F(x) = x is antiderivative for function f(x) = 1.
Why? Because F’(x) = f(x) = 1. The derivative of x is one. No objections.)
The term "prototype" in common parlance means "ancestress", "parent", "ancestor". We immediately remember our dearest and loved one.) And the search for the antiderivative itself is the restoration of the original function by its known derivative. In other words, this action inverse of differentiation. That's all! This fascinating process itself is also called quite scientifically - integration. But about integrals- Later. Patience, friends!)
Remember:
Integration is a mathematical operation on a function (like differentiation).
Integration is the inverse operation of differentiation.
The antiderivative is the result of integration.
Now let's complicate the task. Let us now find an antiderivative for the function f(x) = x. That is, we will find such a function F(x) , to its derivative would be equal to X:
F'(x) = x
Anyone who is familiar with derivatives will probably come to mind something like:
(x 2)’ = 2x.
Well, respect and respect to those who remember the table of derivatives!) That's right. But there is one problem. Our original function f(x) = x, A (x 2)’ = 2 x. Two X. And after differentiation we should get just x. Not okay. But…
You and I are a learned people. We received our certificates.) And from school we know that both sides of any equality can be multiplied and divided by the same number (except zero, of course)! That's it arranged. So let’s realize this opportunity for our own benefit.)
We want a pure X to remain on the right, right? But the two gets in the way... So we take the ratio for the derivative (x 2)’ = 2x and divide both parts of it to this very two:
So, something is already becoming clearer. Go ahead. We know that any constant can be take the derivative out of the sign. Like this:
All formulas in mathematics work both from left to right and vice versa - from right to left. This means that, with the same success, any constant can be insert under the derivative sign:
In our case, we hide the two in the denominator (or, which is the same thing, the coefficient 1/2) under the derivative sign:
And now attentively Let's take a closer look at our recording. What do we see? We see an equality stating that the derivative of something(This something- in parentheses) equals X.
The resulting equality just means that the desired antiderivative for the function f(x) = x serves function F(x) = x 2 /2 . The one in brackets under the stroke. Directly in the meaning of the antiderivative.) Well, let's check the result. Let's find the derivative:
Great! The original function is obtained f(x) = x. What they danced from is what they returned to. This means that our antiderivative was found correctly.)
And if f(x) = x2? What is its antiderivative equal to? No problem! You and I know (again, from the rules of differentiation) that:
3x 2 = (x 3)’
AND, that is,
Got it? Now we, imperceptibly for ourselves, have learned to count antiderivatives for any power function f(x)=x n. In the mind.) Take the initial indicator n, increase it by one, and as compensation divide the entire structure by n+1:
The resulting formula, by the way, is correct not only for natural indicator degrees n, but also for any other – negative, fractional. This makes it easy to find antiderivatives from simple ones fractions And roots
For example:
Naturally, n ≠ -1 , otherwise the denominator of the formula turns out to be zero, and the formula loses its meaning.) About this a special case n = -1 a little bit later.)
What is an indefinite integral? Table of integrals.
Let's say what the derivative of the function is equal to F(x) = x? Well, one, one - I hear dissatisfied answers... That's right. Unit. But... For the function G(x) = x+1 derivative will also be equal to one:
Also, the derivative will be equal to unity for the function x+1234 , and for the function x-10 , and for any other function of the form x+C , Where WITH – any constant. Because the derivative of any constant is equal to zero, and adding/subtracting zero makes no one feel cold or hot.)
This results in ambiguity. It turns out that for the function f(x) = 1 serves as a prototype not just a function F(x) = x , but also a function F 1 (x) = x+1234 and function F 2 (x) = x-10 and so on!
Yes. Exactly so.) For every ( continuous on the interval) of a function there is not just one antiderivative, but infinitely many - the whole family! Not just one mom or dad, but a whole family tree, yeah.)
But! All our primitive relatives have one thing in common: important property. That's why they are relatives.) The property is so important that in the process of analyzing integration techniques we will remember it more than once. And we will remember it for a long time.)
Here it is, this property:
Any two antiderivatives F 1 (x) AndF 2 (x) from the same functionf(x) differ by a constant:
F 1 (x) - F 2 (x) = S.
If anyone is interested in proof, study the literature or lecture notes.) Okay, so be it, I’ll prove it. Fortunately, the proof here is elementary, in one step. Let's take equality
F 1 (x) - F 2 (x) = C
And Let's differentiate both of its parts. That is, we just stupidly add strokes:
That's all. As they say, CHT. :)
What does this property mean? And about the fact that two different antiderivatives from the same function f(x) cannot differ by some kind of expression with an X . Only strictly on a constant! In other words, if we have some kind of schedule one of the original(let it be F(x)), then the graphs everyone else Our antiderivatives are constructed by parallel transfer of the graph F(x) along the y-axis.
Let's see what it looks like using the example function f(x) = x. All its primitives, as we already know, have the general form F(x) = x 2 /2+C . In the picture it looks like infinite number of parabolas, obtained from the “main” parabola y = x 2 /2 by shifting up or down along the OY axis depending on the value of the constant WITH.
Remember the school graphing of a function y=f(x)+a schedule shift y=f(x) by “a” units along the Y-axis?) Same thing here.)
Moreover, pay attention: our parabolas do not intersect anywhere! It is natural. After all, two various functions y 1 (x) and y 2 (x) will inevitably correspond two different meanings constants – C 1 And C 2.
Therefore, the equation y 1 (x) = y 2 (x) never has solutions:
C 1 = C 2
x ∊ ∅ , because C 1 ≠ C2
And now we are gradually approaching the second cornerstone concept of integral calculus. As we have just established, for any function f(x) there is an infinite set of antiderivatives F(x) + C, differing from each other by a constant. This most infinite set also has its own special name.) Well, please love and favor!
What is an indefinite integral?
The set of all antiderivatives for a function f(x) is called indefinite integral from functionf(x).
That's the whole definition.)
"Uncertain" - because the set of all antiderivatives for the same function endlessly. Too many different options.)
"Integral" – we will get acquainted with a detailed decoding of this brutal word in the next large section dedicated to definite integrals . For now, in rough form, we will consider something as an integral general, united, whole. And by integration - Union, generalization, in this case, the transition from the particular (derivative) to the general (antiderivative). Something like that.
The indefinite integral is denoted like this:
It reads the same way as it is written: integral ef from x de x. Or integral from ef from x de x. Well, you understand.)
Now let's look at the notation.
∫ - integral icon. The meaning is the same as the prime for a derivative.)
d - icondifferential. Let's not be afraid! Why it is needed there is a little lower.
f(x) - integrand(through "s").
f(x)dx - integrand expression. Or, roughly speaking, the “filling” of the integral.
According to the meaning of the indefinite integral,
Here F(x)- the same one antiderivative for function f(x) which we somehow found it ourselves. How exactly they found it is not the point. For example, we found that F(x) = x 2 /2 For f(x)=x.
"WITH" - arbitrary constant. Or, more scientifically, integral constant. Or integration constant. Everything is one.)
Now let’s return to our very first examples of finding an antiderivative. In terms of the indefinite integral, we can now safely write:
What is an integral constant and why is it needed?
The question is very interesting. And very (VERY!) important. From the entire infinite set of antiderivatives, the integral constant singles out the line which passes through given point.
What's the point? From the initial infinite set of antiderivatives (i.e. indefinite integral) you need to select the curve that will pass through the given point. With some specific coordinates. Such a task always and everywhere occurs during initial acquaintance with integrals. Both at school and at university.
Typical problem:
Among the set of all antiderivatives of the function f=x, select the one that passes through the point (2;2).
We begin to think with our heads... The set of all primitives means that first we must integrate our original function. That is, x(x). We did this a little higher and got the following answer:
Now let’s figure out what exactly we got. We got not just one function, but a whole family of functions. Which ones? Vida y=x 2 /2+C . Dependent on the value of the constant C. And it is this value of the constant that we now have to “catch”.) Well, let’s start catching?)
Our fishing rod - family of curves (parabolas) y=x 2 /2+C.
Constants - these are fish. Lots and lots. But each has its own hook and bait.)
What is the bait? Right! Our point is (-2;2).
So we substitute the coordinates of our point into the general form of antiderivatives! We get:
y(2) = 2
It's easy to find from here C=0.
What does this mean? This means that from the entire infinite set of parabolas of the formy=x 2 /2+Conly parabola with constant C=0 suits us! Namely:y=x 2 /2. And only her. Only this parabola will pass through the point we need (-2; 2). And inall other parabolas from our family pass through this point they won't be anymore. Through some other points of the plane - yes, but through the point (2; 2) - no longer. Got it?
For clarity, here are two pictures - the entire family of parabolas (i.e. an indefinite integral) and some specific parabola, corresponding specific value of the constant and passing through specific point:
You see how important it is to take into account the constant WITH upon integration! So don’t neglect this letter “C” and don’t forget to add it to the final answer.
Now let’s figure out why the symbol hangs out everywhere inside the integrals dx . Students often forget about it... And this, by the way, is also a mistake! And quite rude. The whole point is that integration is the inverse operation of differentiation. And what exactly is result of differentiation? Derivative? True, but not entirely. Differential!
In our case, for the function f(x) the differential of its antiderivative F(x), will:
For those who do not understand this chain, urgently repeat the definition and meaning of the differential and how exactly it is revealed! Otherwise, you will slow down mercilessly in the integrals...
Let me remind you, in the crudest philistine form, that the differential of any function f(x) is simply the product f'(x)dx. That's all! Take the derivative and multiply it to the differential argument(i.e. dx). That is, any differential, in essence, comes down to calculating the usual derivative.
Therefore, strictly speaking, the integral is not “taken” from functions f(x), as is commonly believed, and from differential f(x)dx! But, in a simplified version, it is customary to say that "the integral is taken from the function". Or: "The function f is integrated(x)". It is the same. And we will speak exactly the same way. But about the badge dx Let's not forget! :)
And now I’ll tell you how not to forget it when recording. First imagine that you are calculating the ordinary derivative with respect to the variable x. How do you usually write it?
Like this: f’(x), y’(x), y’ x. Or more solidly, through the differential ratio: dy/dx. All these records show us that the derivative is taken precisely with respect to X. And not by “igrek”, “te” or some other variable.)
The same goes for integrals. Record ∫ f(x)dx U.S. too as if shows that the integration is carried out precisely by variable x. Of course, this is all very simplified and crude, but it’s understandable, I hope. And the chances forget attribute omnipresence dx decline sharply.)
So, we figured out what an indefinite integral is. Great.) Now it would be good to learn these same indefinite integrals calculate. Or, simply put, “take.” :) And here two news awaits students - good and not so good. For now, let's start with the good one.)
The news is good. For integrals, as well as for derivatives, there is a table of its own. And all the integrals that we will encounter along the way, even the most terrible and sophisticated ones, we according to certain rules One way or another we will reduce it to these very tabular ones.)
So here she is table of integrals!
Here is such a beautiful table of integrals from the most popular functions. I recommend paying special attention to the group of formulas 1-2 (constant and power function). These are the most commonly used formulas in integrals!
The third group of formulas (trigonometry), as you might guess, is obtained by simply inverting the corresponding formulas for derivatives.
For example:
With the fourth group of formulas (exponential function) everything is similar.
Here are four latest groups formulas (5-8) for us new. Where did they come from and for what merit did these exotic functions suddenly enter the table of basic integrals? Why do these groups of functions stand out so much from other functions?
This is how it happened historically in the process of development integration methods . When we practice taking the widest variety of integrals, you will understand that integrals of the functions listed in the table occur very, very often. So often that mathematicians classified them as tabular ones.) Many other integrals, from more complex constructions, are expressed through them.
Just for fun, you can take one of these terrible formulas and differentiate it. :) For example, the most brutal 7th formula.
Everything is fine. The mathematicians were not deceived. :)
It is advisable to know the table of integrals, as well as the table of derivatives, by heart. In any case, the first four groups of formulas. It's not as difficult as it seems at first glance. Memorize the last four groups (with fractions and roots) Bye not worth it. Anyway, at first you will be confused about where to write the logarithm, where the arctangent, where the arcsine, where 1/a, where 1/2a... There is only one way out - solve more examples. Then the table will gradually be remembered by itself, and doubts will stop gnawing.)
Particularly inquisitive persons, taking a closer look at the table, may ask: where in the table are the integrals of other elementary “school” functions – tangent, logarithm, “arcs”? Let's say why there is an integral from sine in the table, but there is NO, say, integral from tangent tg x? Or there is no integral of the logarithm ln x? From arcsine arcsin x? Why are they worse? But it’s full of some “left-handed” functions - with roots, fractions, squares...
Answer. No worse.) Just the above integrals (from tangent, logarithm, arcsine, etc.) are not tabular . And they occur in practice much less often than those presented in the table. Therefore, know by heart, what they are equal to is not at all necessary. It's enough just to know how are they are calculated.)
What, someone still can’t stand it? So be it, especially for you!
Well, are you going to memorize it? :) Won't you? And don’t.) But don’t worry, we will definitely find all such integrals. In the corresponding lessons. :)
Well, now let's move on to the properties of the indefinite integral. Yes, yes, nothing can be done! A new concept is introduced and some of its properties are immediately considered.
Properties of the indefinite integral.
Now the not so good news.
Unlike differentiation, general standard rules of integration, fair for all occasions, not in mathematics. It is fantastic!
For example, you all know very well (I hope!) that any work any two functions f(x) g(x) is differentiated like this:
(f(x) g(x))’ = f’(x) g(x) + f(x) g’(x).
Any the quotient is differentiated like this:
And any complex function, no matter how complicated it may be, is differentiated like this:
And no matter what functions are hidden under the letters f and g, the general rules will still work and the derivative, one way or another, will be found.
But with integrals this number will no longer work: for a product, a quotient (fraction), as well as a complex function general formulas integration does not exist! There are no standard rules! Or rather, they exist. It was me who offended mathematics in vain.) But, firstly, there are much fewer of them than general rules for differentiation. And secondly, most of the integration methods that we will talk about in the following lessons are very, very specific. And they are valid only for a certain, very limited class of functions. Let's say only for fractional rational functions. Or some others.
And some integrals, although they exist in nature, are not expressed at all through elementary “school” functions! Yes, yes, and there are plenty of such integrals! :)
That is why integration is a much more time-consuming and painstaking task than differentiation. But this also has its own twist. This activity is creative and very exciting.) And, if you master the table of integrals well and master at least two basic techniques, which we will talk about later ( and ), then you will really like integration. :)
Now let’s get acquainted with the properties of the indefinite integral. There are none at all. Here they are.
The first two properties are completely analogous to the same properties for derivatives and are called linearity properties of the indefinite integral . Everything here is simple and logical: the integral of the sum/difference is equal to the sum/difference of the integrals, and the constant factor can be taken out of the sign of the integral.
But the next three properties are fundamentally new for us. Let's look at them in more detail. They sound in Russian as follows.
Third property
The derivative of the integral is equal to the integrand
Everything is simple, like in a fairy tale. If you integrate a function and then find the derivative of the result back, then... you get the original integrand function. :) This property can always (and should) be used to check the final result of integration. You have calculated the integral - differentiate the answer! We got the integrand function - OK. If we didn’t receive it, it means we messed up somewhere. Look for the error.)
Of course, the answer may result in such brutal and cumbersome functions that there is no desire to differentiate them back, yes. But it’s better, if possible, to try to check yourself. At least in those examples where it is easy.)
Fourth property
The differential of the integral is equal to the integrand .
Nothing special here. The essence is the same, only dx appears at the end. According to the previous property and differential opening rules.
Fifth property
The integral of the differential of some function is equal to the sum of this function and an arbitrary constant .
This is also a very simple property. We will also use it regularly in the process of solving integrals. Especially - in and.
Here they are beneficial features. I’m not going to bore you with their rigorous evidence here. I suggest that those who wish to do this do it themselves. Directly in the sense of derivative and differential. I will prove only the last, fifth property, because it is less obvious.
So we have a statement:
We take out the “stuffing” of our integral and open it, according to the definition of the differential:
Just in case, I remind you that, according to our notation for derivative and antiderivative, F’(x) = f(x) .
Now we insert our result back inside the integral:
Received exactly definition of indefinite integral (may the Russian language forgive me)! :)
That's all.)
Well. With this, I consider our initial acquaintance with the mysterious world of integrals to be complete. For today I propose to wrap things up. We are already armed enough to go on reconnaissance. If not a machine gun, then at least a water pistol with basic properties and a table. :) In the next lesson, the simplest harmless examples of integrals for the direct application of the table and the written properties are waiting for us.
See you!