F x 4x 3 6x 2 antiderivative. Integrals for dummies: how to solve, calculation rules, explanation

One of the operations of differentiation is finding the derivative (differential) and applying it to the study of functions.

Equally important is the inverse problem. If the behavior of a function is known in the vicinity of each point of its definition, then how to restore the function as a whole, i.e. over the entire range of its definition. This problem is the subject of study of the so-called integral calculus.

Integration is the reverse action of differentiation. Or the restoration of the function f(x) from the given derivative f`(x). The Latin word “integro” means restoration.

Example #1.

Let (f(x))' = 3x 2 . Find f(x).

Solution:

Based on the differentiation rule, it is easy to guess that f (x) \u003d x 3, because

(x 3) ' = 3x 2 However, it is easy to see that f (x) is found ambiguously. As f (x) you can take f (x) \u003d x 3 +1 f (x) \u003d x 3 +2 f (x) \u003d x 3 -3, etc.

Because the derivative of each of them is 3x2. (The derivative of the constant is 0). All these functions differ from each other by a constant term. Therefore, the general solution of the problem can be written as f(x)= x 3 +C, where C is any constant real number.

Any of the found functions f(x) is called primitive for the function F`(x) = 3x 2

Definition.

The function F(x) is called antiderivative for the function f(x) on a given interval J, if for all x from this interval F`(x) = f(x). So the function F (x) \u003d x 3 is antiderivative for f (x) \u003d 3x 2 on (- ∞ ; ∞). Since, for all x ~ R, the equality is true: F`(x)=(x 3)`=3x 2

As we have already noticed, this function has an infinite set of antiderivatives.

Example #2.

The function is antiderivative for all on the interval (0; +∞), because for all h from this interval, the equality holds.

The task of integration is to find all of its antiderivatives for a given function. The following assertion plays an important role in solving this problem:

A sign of the constancy of a function. If F "(x) \u003d 0 on some interval I, then the function F is a constant on this interval.

Proof.

We fix some x 0 from the interval I. Then for any number x from such an interval, by virtue of the Lagrange formula, one can specify such a number c between x and x 0 that

F (x) - F (x 0) \u003d F "(c) (x-x 0).

By condition, F’ (c) = 0, since c ∈1, therefore,

F(x) - F(x 0) = 0.

So, for all x from the interval I

i.e. the function F remains constant.

All antiderivative functions f can be written using one formula, which is called general form of antiderivatives for the function f. The following theorem is true ( basic property of primitives):

Theorem. Any antiderivative for the function f on the interval I can be written as

F(x) + C, (1) where F(x) is one of the antiderivatives for the function f(x) on the interval I, and C is an arbitrary constant.

Let us explain this statement, in which two properties of the antiderivative are briefly formulated:

1. whatever number we put in expression (1) instead of C, we get the antiderivative for f on the interval I;
2. whichever antiderivative Ф for f on the interval I is taken, one can choose such a number C that for all x from the interval I the equality

Proof.

1. By condition, the function F is the antiderivative for f on the interval I. Therefore, F "(x) \u003d f (x) for any x∈1, therefore (F (x) + C)" \u003d F "(x) + C" \u003d f(x)+0=f(x), i.e. F(x) + C is the antiderivative for the function f.
2. Let Ф (х) be one of the antiderivatives for the function f on the same interval I, i.e. Ф "(x) = f (х) for all x∈I.

Then (Ф (x) - F (x)) "= Ф" (x) - F '(x) = f (x) - f (x) \u003d 0.

It follows from here. due to the sign of the constancy of the function, that the difference Ф (х) - F (х) is a function that takes some constant value C on the interval I.

Thus, for all x from the interval I, the equality Ф(х) - F(x)=С is true, which was to be proved. The main property of the antiderivative can be given a geometric meaning: graphs of any two antiderivatives for the function f are obtained from each other by parallel translation along the y-axis

Questions for abstracts

The function F(x) is an antiderivative for the function f(x). Find F(1) if f(x)=9x2 - 6x + 1 and F(-1) = 2.

Find all antiderivatives for a function

For the function (x) = cos2 * sin2x, find the antiderivative F(x) if F(0) = 0.

For a function, find the antiderivative whose graph passes through the point

Definition of antiderivative.

An antiderivative function f(x) on the interval (a; b) is such a function F(x) that equality holds for any x from a given interval.

If we take into account the fact that the derivative of the constant C is equal to zero, then the equality . Thus, the function f(x) has a set of antiderivatives F(x)+C , for an arbitrary constant C , and these antiderivatives differ from each other by an arbitrary constant value.

Definition of the indefinite integral.

The whole set of antiderivatives of the function f(x) is called the indefinite integral of this function and is denoted .

The expression is called integrand, and f(x) integrand. The integrand is the differential of the function f(x) .

The action of finding an unknown function by its given differential is called uncertain integration, because the result of integration is not one function F(x) , but the set of its antiderivatives F(x)+C .

Based on the properties of the derivative, one can formulate and prove indefinite integral properties(properties of the antiderivative).

Intermediate equalities of the first and second properties of the indefinite integral are given for clarification.

To prove the third and fourth properties, it suffices to find the derivatives of the right-hand sides of the equalities:

These derivatives are equal to the integrands, which is the proof by virtue of the first property. It is also used in the last transitions.

Thus, the integration problem is the inverse problem of differentiation, and there is a very close connection between these problems:

• the first property allows checking integration. To check the correctness of the integration performed, it is enough to calculate the derivative of the result obtained. If the function obtained as a result of differentiation turns out to be equal to the integrand, then this will mean that the integration has been carried out correctly;
• the second property of the indefinite integral allows us to find its antiderivative from the known differential of a function. The direct calculation of indefinite integrals is based on this property.

Consider an example.

Example.

Find the antiderivative of the function whose value is equal to one at x = 1.

Solution.

We know from differential calculus that (just look at the table of derivatives of the basic elementary functions). In this way, . By the second property . That is, we have a set of antiderivatives. For x = 1 we get the value . By condition, this value must be equal to one, therefore, С = 1. The desired antiderivative will take the form .

Example.

Find the indefinite integral and check the result by differentiation.

Solution.

According to the formula of the sine of a double angle from trigonometry , that's why

Antiderivative function and indefinite integral

Fact 1. Integration is the opposite of differentiation, namely, the restoration of a function from the known derivative of this function. The function restored in this way F(x) is called primitive for function f(x).

Definition 1. Function F(x f(x) on some interval X, if for all values x from this interval the equality F "(x)=f(x), that is, this function f(x) is the derivative of the antiderivative function F(x). .

For example, the function F(x) = sin x is the antiderivative for the function f(x) = cos x on the entire number line, since for any value of x (sin x)" = (cos x) .

Definition 2. Indefinite integral of a function f(x) is the collection of all its antiderivatives. This uses the notation

∫

f(x)dx

,

where is the sign ∫ is called the integral sign, the function f(x) is an integrand, and f(x)dx is the integrand.

Thus, if F(x) is some antiderivative for f(x) , then

∫

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

where C - arbitrary constant (constant).

To understand the meaning of the set of antiderivatives of a function as an indefinite integral, the following analogy is appropriate. Let there be a door (a traditional wooden door). Its function is "to be a door". What is the door made of? From a tree. This means that the set of antiderivatives of the integrand "to be a door", that is, its indefinite integral, is the function "to be a tree + C", where C is a constant, which in this context can denote, for example, a tree species. Just as a door is made of wood with some tools, the derivative of a function is "made" of the antiderivative function with formula that we learned by studying the derivative .

Then the table of functions of common objects and their corresponding primitives ("to be a door" - "to be a tree", "to be a spoon" - "to be a metal", etc.) is similar to the table of basic indefinite integrals, which will be given below. The table of indefinite integrals lists common functions, indicating the antiderivatives from which these functions are "made". As part of the tasks for finding the indefinite integral, such integrands are given that can be integrated directly without special efforts, that is, according to the table of indefinite integrals. In more complex problems, the integrand must first be transformed so that tabular integrals can be used.

Fact 2. Restoring a function as an antiderivative, we must take into account an arbitrary constant (constant) C, and in order not to write a list of antiderivatives with various constants from 1 to infinity, you need to write down a set of antiderivatives with an arbitrary constant C, like this: 5 x³+C. So, an arbitrary constant (constant) is included in the expression of the antiderivative, since the antiderivative can be a function, for example, 5 x³+4 or 5 x³+3 and when differentiating 4 or 3 or any other constant vanishes.

We set the integration problem: for a given function f(x) find such a function F(x), whose derivative is equal to f(x).

Example 1 Find the set of antiderivatives of a function

Solution. For this function, the antiderivative is the function

Function F(x) is called antiderivative for the function f(x) if the derivative F(x) is equal to f(x), or, which is the same thing, the differential F(x) is equal to f(x) dx, i.e.

(2)

Therefore, the function is antiderivative for the function . However, it is not the only antiderivative for . They are also functions

where FROM is an arbitrary constant. This can be verified by differentiation.

Thus, if there is one antiderivative for a function, then for it there is an infinite set of antiderivatives that differ by a constant summand. All antiderivatives for a function are written in the above form. This follows from the following theorem.

Theorem (formal statement of fact 2). If a F(x) is the antiderivative for the function f(x) on some interval X, then any other antiderivative for f(x) on the same interval can be represented as F(x) + C, where FROM is an arbitrary constant.

In the following example, we already turn to the table of integrals, which will be given in paragraph 3, after the properties of the indefinite integral. We do this before familiarizing ourselves with the entire table, so that the essence of the above is clear. And after the table and properties, we will use them in their entirety when integrating.

Example 2 Find sets of antiderivatives:

Solution. We find sets of antiderivative functions from which these functions are "made". When mentioning formulas from the table of integrals, for now, just accept that there are such formulas, and we will study the table of indefinite integrals in full a little further.

1) Applying formula (7) from the table of integrals for n= 3, we get

2) Using formula (10) from the table of integrals for n= 1/3, we have

3) Since

then according to formula (7) at n= -1/4 find

Under the integral sign, they do not write the function itself f, and its product by the differential dx. This is done primarily to indicate which variable the antiderivative is being searched for. For example,

, ;

here in both cases the integrand is equal to , but its indefinite integrals in the considered cases turn out to be different. In the first case, this function is considered as a function of a variable x, and in the second - as a function of z .

The process of finding the indefinite integral of a function is called integrating that function.

The geometric meaning of the indefinite integral

Let it be required to find a curve y=F(x) and we already know that the tangent of the slope of the tangent at each of its points is a given function f(x) abscissa of this point.

According to the geometric meaning of the derivative, the tangent of the slope of the tangent at a given point on the curve y=F(x) equal to the value of the derivative F"(x). So, we need to find such a function F(x), for which F"(x)=f(x). Required function in the task F(x) is derived from f(x). The condition of the problem is satisfied not by one curve, but by a family of curves. y=F(x)- one of these curves, and any other curve can be obtained from it by parallel translation along the axis Oy.

Let's call the graph of the antiderivative function of f(x) integral curve. If a F"(x)=f(x), then the graph of the function y=F(x) is an integral curve.

Fact 3. The indefinite integral is geometrically represented by the family of all integral curves as in the picture below. The distance of each curve from the origin is determined by an arbitrary constant (constant) of integration C.

Properties of the indefinite integral

Fact 4. Theorem 1. The derivative of an indefinite integral is equal to the integrand, and its differential is equal to the integrand.

Fact 5. Theorem 2. The indefinite integral of the differential of a function f(x) is equal to the function f(x) up to a constant term , i.e.

(3)

Theorems 1 and 2 show that differentiation and integration are mutually inverse operations.

Fact 6. Theorem 3. The constant factor in the integrand can be taken out of the sign of the indefinite integral , i.e.

Table of antiderivatives

Definition. The function F(x) on a given interval is called antiderivative for the function f(x) , for all x from this interval, if F"(x)=f(x) .

The operation of finding the antiderivative for a function is called integration. It is the inverse of differentiation.

Theorem. Every function (x) continuous on an interval has an antiderivative on the same interval.

Theorem (the main property of the antiderivative). If on some interval the function F(x) is antiderivative for the function f(x ), then on this interval the antiderivative for f(x) will also be the function F(x)+C , where C is an arbitrary constant.

It follows from this theorem that when f(x) has a primitive function F(x) on a given interval, then these primitives are a set. Giving C arbitrary numerical values, each time we will obtain an antiderivative function.

To find primitives use table of antiderivatives. It is obtained from the table of derivatives.

The concept of an indefinite integral

Definition. The set of all antiderivatives for the function f(x) is called indefinite integral and is denoted.

Here f(x) is called integrand, and f(x) dx - integrand.

Therefore, if F(x) is the antiderivative of f(x) , then .

Properties of the indefinite integral

The concept of a definite integral

Consider a flat figure bounded by a graph that is continuous and non-negative on the segment [a; b] function f(x) , segment [a; b] , and straight lines x=a and x=b .

The resulting figure is called curvilinear trapezoid. Let's calculate its area.

To do this, we split the segment [a; b] into n equal segments. The lengths of each of the segments are equal to Δx.

This is a GeoGebra dynamic drawing.
Red elements can be changed

Rice. 1. The concept of a definite integral

On each segment, we will construct rectangles with heights f(x k-1) (Fig. 1).

The area of ​​each such rectangle is equal to S k = f(x k-1)Δx k .

The area of ​​all such rectangles is .

This amount is called integral sum for the function f(x) .

If n→∞ then the area of ​​the figure constructed in this way will differ less and less from the area of ​​the curvilinear trapezoid.

Definition. The boundary of the integral sum when n→∞ is called definite integral, and is written like this: .

reads: "integral from a to b f from xdx "

The number a is called the lower limit of integration, b is the upper limit of integration, the segment [a; b] is the interval of integration.

Properties of the Definite Integral

Newton-Leibniz formula

The definite integral is closely related to the antiderivative and the indefinite integral Newton-Leibniz formula

.

Using the Integral

Integral calculus is widely used in solving various practical problems. Let's consider some of them.

Calculation of volumes of bodies

Let a function be given that specifies the cross-sectional area of ​​the body depending on some variable S = s(x), x[а; b] . Then the volume of a given body can be found by integrating this function within the appropriate limits.
If we are given a body that is obtained by rotating around the Ox axis of a curvilinear trapezoid bounded by some function f(x), x [a; b] . (Fig. 3). Then the cross-sectional areas can be calculated using the well-known formula S \u003d π f 2 (x). Therefore, the formula for the volume of such a body of revolution

Previously, for a given function, guided by various formulas and rules, we found its derivative. The derivative has numerous applications: it is the speed of movement (or, more generally, the speed of any process); slope of the tangent to the graph of the function; using the derivative, you can investigate the function for monotonicity and extrema; It helps to solve optimization problems.

But along with the problem of finding the speed from a known law of motion, there is also an inverse problem - the problem of restoring the law of motion from a known speed. Let's consider one of these problems.

Example 1 A material point moves along a straight line, the speed of its movement at time t is given by the formula v=gt. Find the law of motion.
Solution. Let s = s(t) be the desired law of motion. It is known that s"(t) = v(t). So, to solve the problem, you need to choose a function s = s(t), whose derivative is equal to gt. It is easy to guess that \(s(t) = \frac(gt^ 2)(2) \) Indeed
\(s"(t) = \left(\frac(gt^2)(2) \right)" = \frac(g)(2)(t^2)" = \frac(g)(2) \ cdot 2t=gt\)
Answer: \(s(t) = \frac(gt^2)(2) \)

We note right away that the example is solved correctly, but incompletely. We got \(s(t) = \frac(gt^2)(2) \). In fact, the problem has infinitely many solutions: any function of the form \(s(t) = \frac(gt^2)(2) + C \), where C is an arbitrary constant, can serve as a law of motion, since \(\left (\frac(gt^2)(2) +C \right)" = gt \)

To make the problem more specific, we had to fix the initial situation: indicate the coordinate of the moving point at some point in time, for example, at t = 0. If, say, s(0) = s 0 , then from the equality s(t) = (gt 2)/2 + C we get: s(0) = 0 + C, i.e. C = s 0 . Now the law of motion is uniquely defined: s(t) = (gt 2)/2 + s 0 .

In mathematics, mutually inverse operations are assigned different names, come up with special notations, for example: squaring (x 2) and taking the square root (\(\sqrt(x) \)), sine (sin x) and arcsine (arcsin x) and etc. The process of finding the derivative with respect to a given function is called differentiation, and the inverse operation, i.e. the process of finding a function by a given derivative, - integration.

The term "derivative" itself can be justified "in a worldly way": the function y \u003d f (x) "produces into the world" a new function y" \u003d f "(x). The function y \u003d f (x) acts as if as a “parent”, but mathematicians, of course, do not call it “parent” or “producer”, they say that this is, in relation to the function y "\u003d f" (x) , the primary image, or antiderivative.

Definition. A function y = F(x) is called an antiderivative for a function y = f(x) on an interval X if \(x \in X \) satisfies the equality F"(x) = f(x)

In practice, the interval X is usually not specified, but implied (as the natural domain of the function).

Let's give examples.
1) The function y \u003d x 2 is an antiderivative for the function y \u003d 2x, since for any x the equality (x 2) "\u003d 2x is true
2) The function y \u003d x 3 is an antiderivative for the function y \u003d 3x 2, since for any x the equality (x 3)" \u003d 3x 2 is true
3) The function y \u003d sin (x) is an antiderivative for the function y \u003d cos (x), since for any x the equality (sin (x)) "= cos (x) is true

When finding antiderivatives, as well as derivatives, not only formulas are used, but also some rules. They are directly related to the corresponding rules for computing derivatives.

We know that the derivative of a sum is equal to the sum of the derivatives. This rule generates a corresponding rule for finding antiderivatives.

Rule 1 The antiderivative of a sum is equal to the sum of antiderivatives.

We know that the constant factor can be taken out of the sign of the derivative. This rule generates a corresponding rule for finding antiderivatives.

Rule 2 If F(x) is an antiderivative for f(x), then kF(x) is an antiderivative for kf(x).

Theorem 1. If y = F(x) is the antiderivative for the function y = f(x), then the antiderivative for the function y = f(kx + m) is the function \(y=\frac(1)(k)F(kx+m) \)

Theorem 2. If y = F(x) is an antiderivative for a function y = f(x) on an interval X, then the function y = f(x) has infinitely many antiderivatives, and they all have the form y = F(x) + C.

Integration methods

Variable replacement method (substitution method)

The substitution integration method consists in introducing a new integration variable (that is, a substitution). In this case, the given integral is reduced to a new integral, which is tabular or reducible to it. There are no general methods for selecting substitutions. The ability to correctly determine the substitution is acquired by practice.
Let it be required to calculate the integral \(\textstyle \int F(x)dx \). Let's make a substitution \(x= \varphi(t) \) where \(\varphi(t) \) is a function that has a continuous derivative.
Then \(dx = \varphi " (t) \cdot dt \) and based on the invariance property of the indefinite integral integration formula, we obtain the substitution integration formula:
\(\int F(x) dx = \int F(\varphi(t)) \cdot \varphi " (t) dt \)

Integration of expressions like \(\textstyle \int \sin^n x \cos^m x dx \)

If m is odd, m > 0, then it is more convenient to make the substitution sin x = t.
If n is odd, n > 0, then it is more convenient to make the substitution cos x = t.
If n and m are even, then it is more convenient to make the substitution tg x = t.

Integration by parts

Integration by parts - applying the following formula for integration:
\(\textstyle \int u \cdot dv = u \cdot v - \int v \cdot du \)
or:
\(\textstyle \int u \cdot v" \cdot dx = u \cdot v - \int v \cdot u" \cdot dx \)

Table of indefinite integrals (antiderivatives) of some functions

$$ \int 0 \cdot dx = C $$ $$ \int 1 \cdot dx = x+C $$ $$ \int x^n dx = \frac(x^(n+1))(n+1 ) +C \;\; (n \neq -1) $$ $$ \int \frac(1)(x) dx = \ln |x| +C $$ $$ \int e^x dx = e^x +C $$ $$ \int a^x dx = \frac(a^x)(\ln a) +C \;\; (a>0, \;\; a \neq 1) $$ $$ \int \cos x dx = \sin x +C $$ $$ \int \sin x dx = -\cos x +C $$ $ $ \int \frac(dx)(\cos^2 x) = \text(tg) x +C $$ $$ \int \frac(dx)(\sin^2 x) = -\text(ctg) x +C $$ $$ \int \frac(dx)(\sqrt(1-x^2)) = \text(arcsin) x +C $$ $$ \int \frac(dx)(1+x^2 ) = \text(arctg) x +C $$ $$ \int \text(ch) x dx = \text(sh) x +C $$ $$ \int \text(sh) x dx = \text(ch ) x + C $$