Antiderivative of function 3 x. Lecture "An antiderivative. The concept of an antiderivative. The main property of an antiderivative function" (11th grade). Problems to solve independently

Definition 1. Function F(x) is called antiderivative of the function f(x) on a certain interval, if at each point of this interval the function F(x) is differentiable and the equality holds F "(x) = f(x).

Example 1. Function F(x) = sin x is antiderivative function f(x) = cos x on an infinite interval (– ¥; +¥), since

F’(x) = (sin x) " = cos x = f(x) For x Î (– ¥;+¥).

It is easy to verify that the functions F 1 (x) = sin x+ 5 and F 2 (x) = sin x– 10 are also antiderivatives of the function f(x) = cos x for all (– ¥;+¥), i.e. if for function f(x) there is an antiderivative of the function on some interval, then it is not unique. Let us prove that the set of all antiderivatives for a given function f(x) is a set that is given by the formula F(x) + C, Where C– any constant.

Theorem 1 (on the general form of the antiderivative). Let F(x) is one of the antiderivatives for the function f(x) on the interval ( a;b). Then any other antiderivative for the function f(x) on the interval ( a;b) is presented in the form F(x) + C, Where C– a certain number.

Proof. First, let's check that F(x) + C is also an antiderivative of the function f(x) on the interval ( a;b).

According to the conditions of the theorem F(x) on the interval ( a;b f(x), so the equality holds:

F "(x) = f(x) for any xÎ ( a;b).

Because WITH– some number, then

(F(x) + WITH) " = F"(x)+WITH" = F "(x) + 0 = f(x).

This implies: ( F(x) + C)" = f(x) for any xÎ ( a;b), which means F(x) + WITH on the interval ( a;b) is an antiderivative of the function f(x).

Secondly, let's check that if F(x) and F( x) – two antiderivatives for the function f(x) on the interval ( a;b), then they differ from each other by a constant amount, i.e. F(x) – F( x) = const.

Let us denote j( x) = F(x) – F( x). Since by the assumption of the function F(x) and F( x) antiderivatives on the interval ( a;b) for function f(x), then the following equalities hold: F "(x) = f(x) and F"( x) = f(x) for any xÎ ( a;b). Therefore, j"( x) = F "(x) – F" ( x) = f(x) – f(x) = 0 for any xÎ ( a;b).

Function j( x) is continuous and differentiable at xÎ ( a;b). This means that on any segment [ x 1 ; x 2 ] М ( a; b) function j( x) satisfies Lagrange’s theorem: there exists a point О( x 1 ; x 2), for which the equality holds:

j( x 2) – j( x 1) = j" ()× ( x 2 – x 1) = 0×( x 2 – x 1) = 0

Þ j( x 2) – j( x 1) = 0 Þ j( x 2) = j( x 1) Þ j( x) = const.

Means, F(x) – F( x) = const.

So, we got that if one antiderivative is known F(x) for function f(x) on the interval ( a;b), then any other antiderivative can be represented in the form F(x) + WITH, Where WITH– arbitrary constant value. This form of writing antiderivatives is called general view antiderivative.

The concept of an indefinite integral

Definition 2. The set of all antiderivatives for a given function f(x) on the interval ( a;b) is called indefinite integral of the function f(x) on this interval and is denoted by the symbol:

In the designation the sign is called integral sign, – integrand, – integrand function, – integration variable.

Theorem 2. If the function f(x) is continuous on the interval ( a;b), then it has on the interval ( a;b) antiderivative and indefinite integral.

Comment. The operation of finding the indefinite integral of a given function f(x) on a certain interval is called integration of the function f(x).

Properties of the indefinite integral

From the definitions of antiderivative F(x) and the indefinite integral of this function f(x) on a certain interval the properties of the indefinite integral follow:

1. .

2. .

3. , Where WITH– arbitrary constant.

4. , Where k= const.

Comment. All of the above properties are true provided that the integrals appearing in them are considered on the same interval and exist.

Table of basic indefinite integrals

The action of integration is the opposite of the action of differentiation, i.e. for a given derivative function f(x) it is necessary to restore the initial function F(x). Then from Definition 2 and the table of derivatives (see §4, paragraph 3, p. 24) it turns out table of basic integrals.

3. .

4. .

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Antiderivative for function

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Antiderivative Indefinite integral

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Integration. 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...

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Antiderivative

Definition of an antiderivative function

• Function y=F(x) is called the antiderivative of the function y=f(x) at a given interval X, if for everyone X ∈X equality holds: F′(x) = f(x)

Can be read in two ways:

1. f derivative of a function F
2. F antiderivative of a function f

Property of antiderivatives

• If F(x)- antiderivative of a function f(x) on a given interval, then the function f(x) has infinitely many antiderivatives, and all these antiderivatives can be written in the form F(x) + C, where C is an arbitrary constant.

Geometric interpretation

• Graphs of all antiderivatives of a given function f(x) are obtained from the graph of any one antiderivative by parallel translations along the O axis at.

Rules for calculating antiderivatives

1. The antiderivative of the sum is equal to the sum of the antiderivatives. If F(x)- antiderivative for f(x), and G(x) is an antiderivative for g(x), That F(x) + G(x)- antiderivative for f(x) + g(x).
2. The constant factor can be taken out of the sign of the derivative. If F(x)- antiderivative for f(x), And k- constant, then k·F(x)- antiderivative for k f(x).
3. If F(x)- antiderivative for f(x), And k, b- constant, and k ≠ 0, That 1/k F(kx + b)- antiderivative for f(kx + b).

Remember!

Any function F(x) = x 2 + C , where C is an arbitrary constant, and only such a function is an antiderivative for the function f(x) = 2x.

• For example:

  F"(x) = (x 2 + 1)" = 2x = f(x);

  f(x) = 2x, because F"(x) = (x 2 – 1)" = 2x = f(x);

  f(x) = 2x, because F"(x) = (x 2 –3)" = 2x = f(x);

Relationship between the graphs of a function and its antiderivative:

1. If the graph of a function f(x)>0 F(x) increases over this interval.
2. If the graph of a function f(x)<0 on the interval, then the graph of its antiderivative F(x) decreases over this interval.
3. If f(x)=0, then the graph of its antiderivative F(x) at this point changes from increasing to decreasing (or vice versa).

To denote the antiderivative, the sign of the indefinite integral is used, that is, the integral without indicating the limits of integration.

Indefinite integral

Definition:

• The indefinite integral of the function f(x) is the expression F(x) + C, that is, the set of all antiderivatives of a given function f(x). The indefinite integral is denoted as follows: \int f(x) dx = F(x) + C

• f(x)- called the integrand function;
• f(x)dx- called the integrand;
• x- called the variable of integration;
• F(x)- one of the antiderivatives of the function f(x);
• WITH- arbitrary constant.

Properties of the indefinite integral

1. The derivative of the indefinite integral is equal to the integrand: (\int f(x) dx)\prime= f(x) .
2. The constant factor of the integrand can be taken out of the integral sign: \int k \cdot f(x) dx = k \cdot \int f(x) dx.
3. The integral of the sum (difference) of functions is equal to the sum (difference) of the integrals of these functions: \int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx.
4. If k, b are constants, and k ≠ 0, then \int f(kx + b) dx = \frac(1)(k) \cdot F(kx + b) + C.

Table of antiderivatives and indefinite integrals

Function f(x)	Antiderivative F(x) + C	Indefinite integrals \int f(x) dx = F(x) + C
0	C	\int 0 dx = C
f(x) = k	F(x) = kx + C	\int kdx = kx + C
f(x) = x^m, m\not =-1	F(x) = \frac(x^(m+1))(m+1) + C	\int x(^m)dx = \frac(x^(m+1))(m+1) + C
f(x) = \frac(1)(x)	F(x) = l n \lvert x \rvert + C	\int \frac(dx)(x) = l n \lvert x \rvert + C
f(x) = e^x	F(x) = e^x + C	\int e(^x )dx = e^x + C
f(x) = a^x	F(x) = \frac(a^x)(l na) + C	\int a(^x )dx = \frac(a^x)(l na) + C
f(x) = \sin x	F(x) = -\cos x + C	\int \sin x dx = -\cos x + C
f(x) = \cos x	F(x) =\sin x + C	\int \cos x dx = \sin x + C
f(x) = \frac(1)(\sin (^2) x)	F(x) = -\ctg x + C	\int \frac (dx)(\sin (^2) x) = -\ctg x + C
f(x) = \frac(1)(\cos (^2) x)	F(x) = \tg x + C	\int \frac(dx)(\sin (^2) x) = \tg x + C
f(x) = \sqrt(x)	F(x) =\frac(2x \sqrt(x))(3) + C
f(x) =\frac(1)( \sqrt(x))	F(x) =2\sqrt(x) + C
f(x) =\frac(1)( \sqrt(1-x^2))	F(x)=\arcsin x + C	\int \frac(dx)( \sqrt(1-x^2))=\arcsin x + C
f(x) =\frac(1)( \sqrt(1+x^2))	F(x)=\arctg x + C	\int \frac(dx)( \sqrt(1+x^2))=\arctg x + C
f(x)=\frac(1)( \sqrt(a^2-x^2))	F(x)=\arcsin\frac (x)(a)+ C	\int \frac(dx)( \sqrt(a^2-x^2)) =\arcsin \frac (x)(a)+ C
f(x)=\frac(1)( \sqrt(a^2+x^2))	F(x)=\arctg \frac (x)(a)+ C	\int \frac(dx)( \sqrt(a^2+x^2)) = \frac (1)(a) \arctg \frac (x)(a)+ C
f(x) =\frac(1)( 1+x^2)	F(x)=\arctg + C	\int \frac(dx)( 1+x^2)=\arctg + C
f(x)=\frac(1)( \sqrt(x^2-a^2)) (a \not= 0)	F(x)=\frac(1)(2a)l n \lvert \frac (x-a)(x+a) \rvert + C	\int \frac(dx)( \sqrt(x^2-a^2))=\frac(1)(2a)l n \lvert \frac (x-a)(x+a) \rvert + C
f(x)=\tg x	F(x)= - l n \lvert \cos x \rvert + C	\int \tg x dx =- l n \lvert \cos x \rvert + C
f(x)=\ctg x	F(x)= l n \lvert \sin x \rvert + C	\int \ctg x dx = l n \lvert \sin x \rvert + C
f(x)=\frac(1)(\sin x)	F(x)= l n \lvert \tg \frac(x)(2) \rvert + C	\int \frac (dx)(\sin x) = l n \lvert \tg \frac(x)(2) \rvert + C
f(x)=\frac(1)(\cos x)	F(x)= l n \lvert \tg (\frac(x)(2) +\frac(\pi)(4)) \rvert + C	\int \frac (dx)(\cos x) = l n \lvert \tg (\frac(x)(2) +\frac(\pi)(4)) \rvert + C

Newton–Leibniz formula

Let f(x) this function F its arbitrary antiderivative.

\int_(a)^(b) f(x) dx =F(x)|_(a)^(b)= F(b) - F(a)

Where F(x)- antiderivative for f(x)

That is, the integral of the function f(x) on an interval is equal to the difference of antiderivatives at points b And a.

Area of ​​a curved trapezoid

Curvilinear trapezoid is a figure bounded by the graph of a function that is non-negative and continuous on an interval f, Ox axis and straight lines x = a And x = b.

The area of ​​a curved trapezoid is found using the Newton-Leibniz formula:

S= \int_(a)^(b) f(x) dx

Lesson and presentation on the topic: "An antiderivative function. Graph of a function"

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Antiderivative function. Introduction

Guys, you know how to find derivatives of functions using various formulas and rules. Today we will study the inverse operation of calculating the derivative. The concept of derivative is often used in real life. Let me remind you: the derivative is the rate of change of a function at a specific point. Processes involving motion and speed are well described in these terms.

Let's look at this problem: “The speed of an object moving in a straight line is described by the formula $V=gt$. It is required to restore the law of motion.
Solution.
We know the formula well: $S"=v(t)$, where S is the law of motion.
Our task comes down to finding a function $S=S(t)$ whose derivative is equal to $gt$. Looking carefully, you can guess that $S(t)=\frac(g*t^2)(2)$.
Let's check the correctness of the solution to this problem: $S"(t)=(\frac(g*t^2)(2))"=\frac(g)(2)*2t=g*t$.
Knowing the derivative of the function, we found the function itself, that is, we performed the inverse operation.
But it’s worth paying attention to this moment. The solution to our problem requires clarification; if we add any number (constant) to the found function, then the value of the derivative will not change: $S(t)=\frac(g*t^2)(2)+c,c=const$.
$S"(t)=(\frac(g*t^2)(2))"+c"=g*t+0=g*t$.

Guys, pay attention: our problem has an infinite number of solutions!
If the problem does not specify an initial or some other condition, do not forget to add a constant to the solution. For example, our task may specify the position of our body at the very beginning of the movement. Then it is not difficult to calculate the constant; by substituting zero into the resulting equation, we obtain the value of the constant.

What is this operation called?
The inverse operation of differentiation is called integration.
Finding a function from a given derivative – integration.
The function itself will be called an antiderivative, that is, the image from which the derivative of the function was obtained.
It is customary to write the antiderivative with a capital letter $y=F"(x)=f(x)$.

Definition. The function $y=F(x)$ is called the antiderivative of the function $у=f(x)$ on the interval X if for any $хϵХ$ the equality $F’(x)=f(x)$ holds.

Let's make a table of antiderivatives for various functions. It should be printed out as a reminder and memorized.

In our table, no initial conditions were specified. This means that a constant should be added to each expression on the right side of the table. We will clarify this rule later.

Rules for finding antiderivatives

Let's write down a few rules that will help us in finding antiderivatives. They are all similar to the rules of differentiation.

Rule 1. The antiderivative of a sum is equal to the sum of the antiderivatives. $F(x+y)=F(x)+F(y)$.

Example.
Find the antiderivative for the function $y=4x^3+cos(x)$.
Solution.
The antiderivative of the sum is equal to the sum of the antiderivatives, then we need to find the antiderivative for each of the presented functions.
$f(x)=4x^3$ => $F(x)=x^4$.
$f(x)=cos(x)$ => $F(x)=sin(x)$.
Then the antiderivative of the original function will be: $y=x^4+sin(x)$ or any function of the form $y=x^4+sin(x)+C$.

Rule 2. If $F(x)$ is an antiderivative for $f(x)$, then $k*F(x)$ is an antiderivative for the function $k*f(x)$.(We can easily take the coefficient as a function).

Example.
Find antiderivatives of functions:
a) $y=8sin(x)$.
b) $y=-\frac(2)(3)cos(x)$.
c) $y=(3x)^2+4x+5$.
Solution.
a) The antiderivative of $sin(x)$ is minus $cos(x)$. Then the antiderivative of the original function will take the form: $y=-8cos(x)$.

B) The antiderivative of $cos(x)$ is $sin(x)$. Then the antiderivative of the original function will take the form: $y=-\frac(2)(3)sin(x)$.

C) The antiderivative for $x^2$ is $\frac(x^3)(3)$. The antiderivative for x is $\frac(x^2)(2)$. The antiderivative of 1 is x. Then the antiderivative of the original function will take the form: $y=3*\frac(x^3)(3)+4*\frac(x^2)(2)+5*x=x^3+2x^2+5x$ .

Rule 3. If $у=F(x)$ is an antiderivative for the function $y=f(x)$, then the antiderivative for the function $y=f(kx+m)$ is the function $y=\frac(1)(k)* F(kx+m)$.

Example.
Find antiderivatives of the following functions:
a) $y=cos(7x)$.
b) $y=sin(\frac(x)(2))$.
c) $y=(-2x+3)^3$.
d) $y=e^(\frac(2x+1)(5))$.
Solution.
a) The antiderivative of $cos(x)$ is $sin(x)$. Then the antiderivative for the function $y=cos(7x)$ will be the function $y=\frac(1)(7)*sin(7x)=\frac(sin(7x))(7)$.

B) The antiderivative of $sin(x)$ is minus $cos(x)$. Then the antiderivative for the function $y=sin(\frac(x)(2))$ will be the function $y=-\frac(1)(\frac(1)(2))cos(\frac(x)(2) )=-2cos(\frac(x)(2))$.

C) The antiderivative for $x^3$ is $\frac(x^4)(4)$, then the antiderivative of the original function $y=-\frac(1)(2)*\frac(((-2x+3) )^4)(4)=-\frac(((-2x+3))^4)(8)$.

D) Slightly simplify the expression to the power $\frac(2x+1)(5)=\frac(2)(5)x+\frac(1)(5)$.
The antiderivative of an exponential function is the exponential function itself. The antiderivative of the original function will be $y=\frac(1)(\frac(2)(5))e^(\frac(2)(5)x+\frac(1)(5))=\frac(5)( 2)*e^(\frac(2x+1)(5))$.

Theorem. If $y=F(x)$ is an antiderivative for the function $y=f(x)$ on the interval X, then the function $y=f(x)$ has infinitely many antiderivatives, and all of them have the form $y=F( x)+С$.

If in all the examples considered above it was necessary to find the set of all antiderivatives, then the constant C should be added everywhere.
For the function $y=cos(7x)$ all antiderivatives have the form: $y=\frac(sin(7x))(7)+C$.
For the function $y=(-2x+3)^3$ all antiderivatives have the form: $y=-\frac(((-2x+3))^4)(8)+C$.

Example.
Given the law of change in the speed of a body over time $v=-3sin(4t)$, find the law of motion $S=S(t)$ if at the initial moment of time the body had a coordinate equal to 1.75.
Solution.
Since $v=S’(t)$, we need to find the antiderivative for a given speed.
$S=-3*\frac(1)(4)(-cos(4t))+C=\frac(3)(4)cos(4t)+C$.
In this problem it is given additional condition- initial moment of time. This means that $t=0$.
$S(0)=\frac(3)(4)cos(4*0)+C=\frac(7)(4)$.
$\frac(3)(4)cos(0)+C=\frac(7)(4)$.
$\frac(3)(4)*1+C=\frac(7)(4)$.
$C=1$.
Then the law of motion is described by the formula: $S=\frac(3)(4)cos(4t)+1$.

Problems to solve independently

1. Find antiderivatives of functions:
a) $y=-10sin(x)$.
b) $y=\frac(5)(6)cos(x)$.
c) $y=(4x)^5+(3x)^2+5x$.
2. Find antiderivatives of the following functions:
a) $y=cos(\frac(3)(4)x)$.
b) $y=sin(8x)$.
c) $y=((7x+4))^4$.
d) $y=e^(\frac(3x+1)(6))$.
3. According to the given law of change in the speed of a body over time $v=4cos(6t)$, find the law of motion $S=S(t)$ if at the initial moment of time the body had a coordinate equal to 2.

Solving integrals is an easy task, but only for a select few. This article is for those who want to learn to understand integrals, but know nothing or almost nothing about them. Integral... Why is it needed? How to calculate it? What is definite and not definite integral s? If the only use you know of for an integral is to use a crochet hook shaped like an integral icon to get something useful out of hard-to-reach places, then welcome! Find out how to solve integrals and why you can't do without it.

We study the concept of "integral"

Integration was known back in Ancient Egypt. Of course not in modern form, but still. Since then, mathematicians have written many books on this topic. Especially distinguished themselves Newton And Leibniz , but the essence of things has not changed. How to understand integrals from scratch? No way! To understand this topic, you will still need a basic knowledge of the basics of mathematical analysis. We already have information about , necessary for understanding integrals, on our blog.

Indefinite integral

Let us have some function f(x) .

Indefinite integral function f(x) this function is called F(x) , whose derivative is equal to the function f(x) .

In other words, an integral is a derivative in reverse or an antiderivative. By the way, read about how in our article.

The antiderivative exists for everyone continuous functions. Also, a constant sign is often added to the antiderivative, since the derivatives of functions that differ by a constant coincide. The process of finding the integral is called integration.

Simple example:

In order not to constantly calculate antiderivatives elementary functions, it is convenient to summarize them in a table and use ready-made values.

Complete table of integrals for students

Definite integral

When dealing with the concept of an integral, we are dealing with infinitesimal quantities. The integral will help calculate the area of ​​the figure, mass inhomogeneous body, passed at uneven movement path and much more. It should be remembered that an integral is the sum of an infinitely large number of infinitesimal terms.

As an example, imagine a graph of some function. How to find the area of ​​a figure bounded by the graph of a function?

Using an integral! Let us divide the curvilinear trapezoid, limited by the coordinate axes and the graph of the function, into infinitesimal segments. This way the figure will be divided into thin columns. The sum of the areas of the columns will be the area of ​​the trapezoid. But remember that such a calculation will give an approximate result. However, the smaller and narrower the segments, the more accurate the calculation will be. If we reduce them to such an extent that the length tends to zero, then the sum of the areas of the segments will tend to the area of ​​the figure. This is a definite integral, which is written like this:

Points a and b are called limits of integration.

Bari Alibasov and the group "Integral"

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Rules for calculating integrals for dummies

Properties of the indefinite integral

How to solve an indefinite integral? Here we will look at the properties of the indefinite integral, which will be useful when solving examples.

• The derivative of the integral is equal to the integrand:

• The constant can be taken out from under the integral sign:

• The integral of the sum is equal to the sum of the integrals. This is also true for the difference:

Properties of a definite integral

• Linearity:

• The sign of the integral changes if the limits of integration are swapped:

• At any points a, b And With:

We have already found out that a definite integral is the limit of a sum. But how to get a specific value when solving an example? For this there is the Newton-Leibniz formula:

Examples of solving integrals

Below we will look at a few examples of finding indefinite integrals. We suggest you figure out the intricacies of the solution yourself, and if something is unclear, ask questions in the comments.

To reinforce the material, watch a video about how integrals are solved in practice. Don't despair if the integral is not given right away. Contact a professional service for students, and any triple or curved integral over a closed surface will be within your power.