Lesson summary on the topic of indefinite integral. Open lesson in algebra
Class: 11
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Technological map of the algebra lesson Grade 11.
“A person can recognize his abilities only by trying to apply them.”
Seneca the Younger.
Number of hours per section: 10 hours.
Block theme: antiderivative and indefinite integral.
Leading topic of the lesson: formation of knowledge and general educational skills through a system of typical, approximate and multi-level tasks.
Lesson Objectives:
- Educational: to form and consolidate the concept of antiderivative, to find antiderivative functions of different levels.
- Developing: develop mental activity students, based on the operations of analysis, comparisons, generalization, systematization.
- Educational: to form the worldview views of students, to educate from responsibility for the result, a sense of success.
Lesson type: learning new material.
Teaching methods: verbal, verbal-visual, problematic, heuristic.
Forms of study: individual, pair, group, general class.
Means of education: informational, computer, epigraph, Handout.
Expected learning outcomes: student must
- definition of derivative
- antiderivative is defined ambiguously.
- find antiderivative functions in the simplest cases
- check whether the antiderivative for a function on a given time interval.
LESSON STRUCTURE:
- Setting the goal of the lesson (2 min)
- Preparation for learning new materials (3 min)
- Acquaintance with new material (25 min)
- Initial reflection and application of what has been learned (10 min)
- staging homework(2 minutes)
- Summing up the lesson (3 min)
- Reserve assignments.
During the classes
1. Message of the topic, purpose of the lesson, tasks and motivation of educational activities.
On the writing board:
*** Derivative - "produces" into the world new feature. Primitive - the primary image.
2. Actualization of knowledge, systematization of knowledge in comparison.
Differentiation-finding the derivative.
Integration is the restoration of a function by a given derivative.
Introduction to new characters:
* oral exercises: instead of points, put some function that satisfies equality. (See presentation) -individual work.
(at this time, 1 student writes differentiation formulas on the board, 2 students - the rules of differentiation).
- self-examination is performed by students. (individual work)
- updating students' knowledge.
3. Learning new material.
A) Reciprocal operations in mathematics.
Teacher: in mathematics there are 2 mutually inverse operations in mathematics. Let's take a look at the comparison.
B) Reciprocal operations in physics.
Two mutually inverse problems are considered in the mechanics section. Finding the speed according to the given equation of motion of a material point (finding the derivative of the function) and finding the equation for the trajectory of motion along well-known formula speed.
Example 1 page 140 - work with a textbook (individual work).
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 with respect to a given derivative, is called integration.
C) A definition of antiderivative is introduced.
Teacher: in order for the task to become more specific, we need to fix the initial situation.
Tasks for the formation of the ability to find the primitive - work in groups. (see presentation)
Tasks for the formation of the ability to prove that the antiderivative is for a function on a given interval - pair work. (see presentation)
4. Primary comprehension and application of what has been learned.
Examples with solutions "Find a mistake" - individual work. (See the presentation)
***perform cross check.
Conclusion: when performing these tasks, it is easy to notice that the antiderivative is determined ambiguously.
5. Setting homework
Read the explanatory text chapter 4 paragraph 20, memorize the definition of 1. primitive, solve No. 20.1 -20.5 (c, d) - a mandatory task for everyone No. 20.6 (b), 20.7 (c, d), 20.8 (b), 20.9 ( b) - 4 examples of choice.
6. Summing up the lesson.
During the frontal survey, together with the students, the results of the lesson are summed up, a conscious understanding of the concept of new material can be in the form of emoticons.
Understood everything, managed everything.
Partially did not understand (a), did not manage to do everything.
7. Reserve tasks.
In case of early completion by the whole class of the tasks proposed above, to ensure employment and development of the most prepared students, it is also planned to use tasks No. 20.6 (a), 20.7 (a), 20.9 (a)
Literature:
- A.G. Mordkovich, P.V. Semenov, Algebra of analysis, profile level, part 1, part 2 problem book, Manvelov S. G. "Fundamentals of the creative development of the lesson."
Topic: Antiderivative and indefinite integral.
Target: students will test and consolidate knowledge and skills on the topic "Anti-derivative and indefinite integral".
Tasks:
educational : learn how to calculate antiderivatives and not definite integrals using properties and formulas;
Educational : will develop critical thinking will be able to observe and analyze mathematical situations;
Educational : students learn to respect other people's opinions, the ability to work in a group.
Expected Result:
They will deepen and systematize theoretical knowledge, develop cognitive interest, thinking, speech, and creativity.
Type of : consolidation lesson
The form: frontal, individual, pair, group.
Teaching methods : partially exploratory, practical.
Methods of knowledge : analysis, logical, comparison.
Equipment: textbook, tables.
Student assessment: self-assessment and self-assessment, observation of children during
lesson time.
During the classes.
Call.
Goal setting:
You and I can plot a quadratic function graph, we can solve quadratic equations and square inequalities, as well as solve systems of linear inequalities.
What do you think the topic of today's lesson will be?
Creation Have a good mood on the lesson. (2-3 min)
Draw the mood:The mood of a person is primarily reflected in the products of his activity: drawings, stories, statements, etc. “My mood”:on a common sheet of drawing paper, with the help of pencils, each child draws his mood in the form of a strip, a cloud, a speck (within a minute).
Then the leaves are passed around. The task of each is to determine the mood of a friend and supplement it, finish it. This continues until the leaves return to their owners.
After that, the resulting drawing is discussed.
III. Frontal survey of students: "Fact or opinion" 17 min
1. Formulate the definition of antiderivative.
2. Which of the functionsare antiderivatives for the function
3. Prove that the functionis the antiderivative of the functionon the interval (0;∞).
4. Formulate the main property of the antiderivative. How is this property interpreted geometrically?
5. For functionfind the antiderivative whose graph passes through the point. (Answer:F( x) = tgx + 2.)
6. Formulate the rules for finding the antiderivative.
7. Formulate a theorem on the area of a curvilinear trapezoid.
8. Write down the Newton-Leibniz formula.
9. What is geometric sense integral?
10. Give examples of the application of the integral.
11. Feedback: "Plus-minus-interesting"
IV. Individual-pair work with peer review: 10 min
Solve #5,6,7
V. Practical work: solve in a notebook. 10 min
Solve #8-10
VI. Lesson results. Grading (OdO, OO). 2 minutes
VII. Homework: p. 1 No. 11,12 1 min
VIII. Reflection: 2 min
Lesson:
Attracted me to...
Looked interesting...
Excited…
Made me think...
Got me thinking...
What made the biggest impression on you?
Will the knowledge gained in this lesson be useful to you later in life?
What new did you learn in the lesson?
What do you need to remember?
10. More work to be done
I had a lesson in 11th grade on the topic"The antiderivative and the indefinite integral", this is a lesson on fixing the topic.
Tasks to be solved during the lesson:
learn how to calculate primitive and indefinite integrals using properties and formulas; will develop critical thinking, will be able to observe and analyze mathematical situations; students learn to respect other people's opinions, the ability to work in a group.
After the lesson, I expected the following result:
Students will deepen and systematize theoretical knowledge, develop cognitive interest, thinking, speech, and creativity.
Create conditions for the development of practical and creative thinking. Raising a responsible attitude to educational work, fostering a sense of respect between students to maximize their abilities through group learning
In her lesson, she used frontal, individual, pair, group work.
I planned this lesson in order to reinforce the concept of the antiderivative and the indefinite integral with the students.
I think I did a good job of creating the "Paint the Mood" poster at the beginning of the lesson.The mood of a person, first of all, is reflected in the products of his activity: drawings, stories, statements, etc. “My mood”: whenon a common sheet of drawing paper with the help of pencils, each child draws his mood (within a minute).
Then the paper turns in a circle. The task of each is to determine the mood of a friend and supplement it, finish it. This continues until the picture on the paper returns to its owner.After that, the resulting drawing is discussed. Each child was able to display their mood and start working in the lesson.
At the next stage of the lesson, using the “Fact or Opinion” method, students tried to prove that all concepts on a given topic are a fact, but not their personal opinion. When solving examples on this topic, perception, comprehension and memorization are ensured. Holistic systems of leading knowledge on this topic are being formed.
During the control and self-examination of knowledge, the quality and level of mastery of knowledge are revealed, as well as methods of action, and their correction is provided.
In the structure of the lesson, I included a partial search task. The children solved the problems on their own. We checked ourselves in the group. Received individual advice. I am constantly looking for new techniques and methods of working with children. Ideally, I would like each child to plan his own activities in the lesson and after it, answer the questions: do I want to reach certain heights or not, do I need education at high level or not. Using the example of this lesson, I tried to show that the child himself can determine both the topic and the course of the lesson.That he himself can adjust his activities and the activities of the teacher in such a way that the lesson and additional classes meet his needs.
When choosing one or another type of tasks, I took into account the purpose of the lesson, the content and difficulties. educational material, type of lesson, ways and methods of teaching, age and psychological features students.
In the traditional system of education, when the teacher presents ready-made knowledge, and the students passively assimilate it, the question of reflection is usually not raised.
I think that the work turned out especially well when compiling the reflection “What I learned (a) in the lesson ...”. This task aroused particular interest and helpedunderstand how best to organize this work in the next lesson.
I think that self-assessment and mutual assessment did not work out, the students overestimated their own and their comrades' marks.
Analyzing the lesson, I realized that the students were well aware of the meaning of formulas and their application in solving and learned to use different strategies at different stages of the lesson.
I want to conduct the next lesson on the Six Hats strategy and conduct the Butterfly reflection, which will allow everyoneexpress your opinion, write it down.
Lesson topic: "Anti-derivative and integral" Grade 11 (review)
Lesson type: lesson of assessment and correction of knowledge; repetition, generalization, formation of knowledge, skills.
Lesson motto : It's not a shame not to know, it's a shame not to learn.
Lesson Objectives:
- Tutorials: repeat theoretical material; to work out the skills of finding antiderivatives, calculating integrals and areas of curvilinear trapezoids.
- Developing: develop independent thinking skills, intellectual skills (analysis, synthesis, comparison, comparison), attention, memory.
- Educational: education of the mathematical culture of students, increasing interest in the material being studied, preparing for the UNT.
Lesson outline plan.
II. Updating the basic knowledge of students.
1.Oral work with the class to repeat definitions and properties:
1. What is called a curvilinear trapezoid?
2. What is the antiderivative for the function f(x)=x2.
3. What is the sign of function constancy?
4. What is called the antiderivative F(x) for the function f(x) on xI?
5. What is the antiderivative for the function f(x)=sinx.
6. Is the statement true: "The antiderivative of the sum of functions is equal to the sum of their antiderivatives"?
7. What is the main property of the antiderivative?
8. What is the antiderivative for the function f(x)=.
9. Is the statement true: “The antiderivative of the product of functions is equal to the product of their
Primitives?
10. What is called an indefinite integral?
11. What is called a definite integral?
12. Name a few examples of the use of a definite integral in geometry and physics.
Answers
1. A figure bounded by the graphs of functions y=f(x), y=0, x=a, x=b is called a curvilinear trapezoid.
2. F(x)=x3/3+С.
3. If F`(x0)=0 on some interval, then the function F(x) is constant on this interval.
4. The function F(x) is called antiderivative for the function f(x) on a given interval, if for all x from this interval F`(x)=f(x).
5. F(x)= - cosx+C.
6. Yes, that's right. This is one of the properties of primitives.
7. Any antiderivative for a function f on a given interval can be written as
F(x)+C, where F(x) is one of the antiderivatives for the function f(x) on a given interval, and C is
Arbitrary constant.
9. No, not true. There is no such property of primitives.
10. If the function y \u003d f (x) has an antiderivative y \u003d F (x) on a given interval, then the set of all antiderivatives y \u003d F (x) + C is called the indefinite integral of the function y \u003d f (x).
11. The difference between the values of the antiderivative function at points b and a for the function y \u003d f (x) on the interval [ a ; b ] is called the definite integral of the function f(x) on the interval [ a; b] .
12.. Calculation of the area of a curvilinear trapezoid, volumes of bodies and calculation of the speed of a body in a certain period of time.
Application of the integral. (Additionally write in notebooks)
Quantities
Derivative calculation
Integral calculation
s - displacement,
A - acceleration
A(t) =
A - work,
F - strength,
N - power
F(x) = A"(x)
N(t) = A"(t)
m is the mass of a thin rod,
Line Density
(x) = m"(x)
q - electric charge,
I - current strength
I(t) = q(t)
Q is the amount of heat
C - heat capacity
c(t) = Q"(t)
Rules for computing antiderivatives
- If F is an antiderivative for f, and G is an antiderivative for g, then F+G is an antiderivative for f+g.
If F is the antiderivative of f and k is a constant, then kF is the antiderivative of kf.
If F(x) is an antiderivative for f(x), ak, b are constants, and k0, that is, there is an antiderivative for f(kx+b).
^ 4) - Newton-Leibniz formula.
5) The area S of the figure bounded by the straight lines x-a, x=b and the graphs of continuous functions on the interval and such that for all x is calculated by the formula
6) The volumes of bodies formed by the rotation of a curvilinear trapezoid bounded by a curve y = f (x), the Ox axis and two straight lines x = a and x = b around the axes Ox and Oy, are calculated respectively by the formulas:
Find the indefinite integral:(orally)
1.
2.
3.
4.
5.
6.
7.
Answers:
1.
2.
3.
4.
5.
6.
7.
III Solving tasks with a class
1. Calculate the definite integral: (in notebooks, one student on the board)
Tasks for drawings with solutions:
№ 1. Find the area of a curvilinear trapezoid bounded by lines y= x3, y=0, x=-3, x=1.
Solution.
-∫ x3 dx + ∫ x3 dx = - (x4/4) | + (x4 /4) | = (-3)4/4 + 1/4 = 82/4 = 20.5
№3. Calculate the area of the figure bounded by the lines y=x3+1, y=0, x=0
№ 5.Calculate the area of \u200b\u200bthe figure bounded by the lines y \u003d 4 -x2, y \u003d 0,
Solution. First, let's plot a graph to determine the limits of integration. The figure consists of two identical pieces. Calculate the area of the part to the right of the y-axis and double it.
№ 4.Calculate the area of the figure bounded by the lines y=1+2sin x, y=0, x=0, x=n/2
F(x) = x - 2cosx; S = F(p/2) - F(0) = p/2 -2cos p/2 - (0 - 2cos0) = p/2 + 2
Calculate the area of curvilinear trapezoids bounded by graphs of lines known to you.
3. Calculate the areas of the shaded figures from the figures ( independent work in pairs)
Task: Calculate the area of the shaded figure
Task: Calculate the area of the shaded figure
III The results of the lesson.
a) reflection: -What conclusions did you draw from the lesson for yourself?
Is there something for everyone to work on on their own?
Was the lesson helpful for you?
b) analysis of student work
c) At home: repeat the properties of all the formulas of antiderivatives, the formulas for finding the area of a curvilinear trapezoid, the volumes of bodies of revolution. No. 136 (Shynybekov)
OPEN LESSON ON THE TOPIC
« GENERAL AND INDETERMINATE INTEGRAL.
PROPERTIES OF THE INDETERMINATE INTEGRAL”.
2 hours.
11a class with in-depth study of mathematics
Problem presentation.
Problem-search learning technologies.
PRIMARY AND INDETERMINATE INTEGRAL.
PROPERTIES OF THE INDETERMINATE INTEGRAL.
THE PURPOSE OF THE LESSON:
Activate mental activity;
Contribute to the assimilation of research methods
- to ensure a more solid assimilation of knowledge.
LESSON OBJECTIVES:
introduce the concept of antiderivative;
prove the theorem on the set of antiderivatives for a given function (using the definition of an antiderivative);
introduce the definition of an indefinite integral;
prove the properties of the indefinite integral;
to develop the skills of using the properties of the indefinite integral.
PRELIMINARY WORK:
repeat the rules and formulas of differentiation
concept of differential.
It is proposed to solve problems. Problems are written on the board.
Students give answers to solve problems 1, 2.
(Updating the experience of solving problems on the use of differential
quoting).
1. The law of motion of the body S(t) , find its instantaneous
speed at any given time.
- V(t) = S(t).
2. Knowing that the amount of electricity flowing
through the conductor is expressed by the formula q (t) = 3t - 2 t,
derive a formula for calculating the current strength in any
point in time t.
- I (t) = 6t - 2.
3 . Knowing the speed of a moving body at each moment of time
me, to find the law of its motion.
Knowing that the strength of the current passing through the conductor in any
determining the amount of electricity passing
through the conductor.
Teacher: Is it possible to solve problems number 3 and 4 using
the funds we have?
(Creating a problem situation).
Student guesses:
- To solve this problem, it is necessary to introduce an operation,
the opposite of differentiation.
The differentiation operation compares to a given
function F (x) its derivative.
F(x) = f(x).
Teacher: What is the task of differentiation?
Students' conclusion:
Based on the given function f (x), find such a function
F (x) whose derivative is f (x) , i.e.
f(x) = F(x) .
This operation is called integration, more precisely
indefinite integration.
The section of mathematics that studies the properties of the operation of integrating functions and its applications to solving problems in physics and geometry is called integral calculus.
Integral calculus _ is a section mathematical analysis, together with differential calculus, it forms the basis of the apparatus of mathematical analysis.
Integral calculus arose from considering a large number problems of natural science and mathematics. The most important of them is the physical problem of determining the distance traveled in a given time along a known, but perhaps variable, speed of movement, and a much more ancient problem - calculating the areas and volumes of geometric figures.
What is the uncertainty of this inverse operation remains to be seen.
Let's introduce a definition. (briefly symbolically written
On the desk).
Definition 1. The function F (x) defined on some interval
ke X, is called the antiderivative for the function given
on the same interval if for all x X
equality
F(x) = f (x) or d F(x) = f (x) dx .
For example. (x) = 2x, this equality implies that the function
x is antiderivative on the whole number line
for the 2x function.
Using the definition of an antiderivative, do the exercise
No. 2 (1,3,6) . Check that the function F is an antiderivative
noah for the function f, if
1) F(x) =
2 cos 2x , f (x) = x - 4 sin 2x .
2) F(x) = tg x - cos 5x, f (x) =
+ 5 sin 5x.
3) F(x) = x sin x +
, f(x) = 4x sinx + x cosx +
.
Solutions to examples are written on the board by students, comments
driving your actions.
Is the function x the only antiderivative
for function 2x?
Students give examples
x + 3; x - 92, etc. ,
Students draw their own conclusions:
Every function has infinitely many antiderivatives.
Any function of the form x + C, where C is some number,
is the antiderivative of x.
The antiderivative theorem is written in a notebook under dictation
teachers.
Theorem. If the function f has an antiderivative on the interval
F, then for any number C the function F + C also
is the antiderivative of f . Other primitives
the function f on X does not.
The proof is carried out by students under the guidance of a teacher.
a) Because F is the antiderivative for f on the interval X, then
F(x) = f(x) for all x X.
Then for x X for any C we have:
(F(x) + C) = f(x) . This means that F (x) + C is also
antiderivative f on X.
b) Let us prove that for other antiderivatives on X the function f
does not have.
Assume that Ф is also an antiderivative for f on X.
Then Ф(x) = f (x) and therefore for all x X we have:
Ф (x) - F (x) = f (x) - f (x) = 0, therefore
Ф - F is constant on X. Let Ф (x) - F (x) = C, then
Ф (x) = F (x) + C, so any antiderivative
function f on X has the form F + C.
Teacher: what is the task of finding all the prototypes
for this function?
The students come up with the following conclusion:
The problem of finding all antiderivatives is solved
finding any one: if such a
different is found, then any other is obtained from it
adding a constant.
The teacher formulates the definition of an indefinite integral.
Definition 2. The set of all antiderivatives of the function f
is called the indefinite integral of this
functions.
Designation.
; - the integral is read.
= F (x) + C, where F is one of the antiderivatives
for f , C runs through the set
real numbers.
f - integrand;
f (x)dx - integrand;
x - integration variable;
C is the constant of integration.
Students study the properties of the indefinite integral from the textbook on their own and write them out in a notebook.
.
Students write solutions in notebooks, working at the blackboard
Lesson topic : Primitive. Indefinite integral and its properties
Lesson Objectives:
Educational:
to acquaint students with the concepts of antiderivative and indefinite integral, the main property of the antiderivative and the rules for finding the antiderivative and indefinite integral.
Developing:
develop skills for independent work,
to activate mental activity, mathematical speech.
Educational:
to cultivate a sense of responsibility for the quality and result of the work performed;
form accountability for the final result.
Type of lesson : messages of new knowledge
Conduct method : verbal, visual, independent work.
Security lesson :
Multimedia equipment and software for displaying presentations and videos;
Handout: a table of simple integrals (at the consolidation stage).
Lesson structure.
1. Organizational moment (2 min.)
Motivation learning activities. (5 min.)
Presentation of new material. (50 min.)
Consolidation of the studied material. (25 min.)
Summing up the lesson. Reflection. (6 min.)
Homework message. (2 min.)
Course progress.
Organizing time. (2 minutes.)
teaching methods
Teaching techniques
The teacher greets students, checks those present in the audience.
The students are getting ready for work. The headman fills out a report. Officers distribute handouts.
Motivation of educational activity. ( 5 minutes.)
teaching methods
Teaching techniques
Topic of today's lesson“Ancient.Indefinite integral and its properties".(Slide 1)
Knowledge on this topic will be used by us in the following lessons when finding certain integrals, areas of flat figures. Much attention is paid to integral calculus in sections higher mathematics in higher educational institutions when solving applied problems.
Our lesson today is the lesson of studying new material, therefore it will be of a theoretical nature. The purpose of the lesson is to form ideas about integral calculus, to understand its essence, to develop skills in finding antiderivatives and indefinite integrals.(Slide 2)
Students write down the date and topic of the lesson.
3. Presentation of new material (50 min)
teaching methods
Teaching techniques
1. We recently covered the topic "Derivatives of some elementary functions". For example:
Function derivativef (x)= X 9 , We know thatf ′(x)= 9x 8 . Now we will consider an example of finding a function whose derivative is known.
Suppose we are given a derivativef ′(x)= 6x 5 . Using knowledge of the derivative, we can determine what is the derivative of the functionf (x)= X 6 . A function that can be determined by its derivative is called antiderivative. (Give a definition of antiderivative. (slide 3))
Definition 1 : Function F ( x ) is called antiderivative for the function f ( x ) on the segment [ a; b], if the equality holds at all points of this segment = f ( x )
Example 1 (slide 4): Let's prove that for anyxϵ(-∞;+∞) functionF ( x )=x 5 -5x f (x)=5 X 4 -5.
Proof: Using the definition of antiderivative, we find the derivative of the function
=(X 5 -5x)′=(x 5 )′-(5х)′=5 X 4 -5.
Example 2 (slide 5): Let's prove that for anyxϵ(-∞;+∞) functionF ( x )= notis the antiderivative for the functionf (x)= .
Prove with students on the blackboard.
We know that finding the derivative is calleddifferentiation . Finding a function by its derivative will be calledintegration. (Slide 6). The goal of integration is to find all antiderivatives of a given function.
For example: (slide 7)
The main property of the antiderivative:
Theorem: IfF ( x ) - one of the antiderivatives for the function f (X) on the interval X, then the set of all antiderivatives of this function is determined by the formula G ( x )= F ( x )+ C where C is a real number.
(Slide 8) table of antiderivatives
Three rules for finding antiderivatives
Rule #1: If a Fthere is an antiderivative for the functionf, a G- original forg, then F+ G- there is a prototype forf+ g.
(F(x) + G(x))' = F'(x) + G'(x) = f + g
Rule #2: If a F- original forf, a kis constant, then the functionkF- original forkf.
(kF)’ = kF’ = kf
Rule #3: If a F- original forf, a k and b are constants (), then the function
antiderivative forf(kx+ b).
The history of the concept of an integral is closely connected with the problems of finding quadratures. Problems about the quadrature of one or another flat figure of mathematics Ancient Greece and Rome called the problems that we now refer to the problems of calculating areas. Many significant achievements of the mathematicians of ancient Greece in solving such problems are associated with the use of the exhaustion method proposed by Eudoxus of Cnidus. With this method, Eudoxus proved:
1. The areas of two circles are related as the squares of their diameters.
2. The volume of a cone is equal to 1/3 of the volume of a cylinder having the same height and base.
The method of Eudoxus was perfected by Archimedes and the following things were proven:
1. Derivation of the formula for the area of a circle.
2. The volume of the sphere is 2/3 of the volume of the cylinder.
All achievements have been proven by great mathematicians using integrals.
Let us return to Theorem 1 and derive a new definition.
Definition 2 : Expression F ( x ) + C , where C - an arbitrary constant, called the indefinite integral and denoted by the symbol
From the definition we have:
(1)
Indefinite integral of a functionf(x), thus, is the set of all antiderivative functions forf(x) .
In equality (1), the functionf(x) is called integrand , and the expression f(x) dx– integrand , variable x – integration variable , term C - integration constant .
Integration is the inverse of differentiation. In order to check whether the integration is correct, it suffices to differentiate the result and obtain the integrand.
Properties of the indefinite integral.
Based on the definition of an antiderivative, it is easy to prove the followingproperties of the indefinite integral
The indefinite integral of the differential of some function is equal to this function plus an arbitrary constant
The indefinite integral of the algebraic sum of two or more functions is equal to the algebraic sum of their integrals
The constant factor can be taken out of the integral sign, that is, ifa= const, then
The students record the lecture using the handout and teacher's explanations. When proving the properties of antiderivatives and integrals, they use knowledge on the topic of differentiation.
4. Table of simple integrals
1. ,( n -1) 2.
3. 4.
5. 6.
The integrals contained in this table are calledtabular . Note special case formula 1:
Here is another obvious formula: