11th grade Orlova E.V.
"Antiderivative and indefinite integral"
SLIDE 1
Lesson objectives:
Educational : form and consolidate the concept of an antiderivative, find antiderivative functions of different levels.
Developmental: develop the mental activity of students based on the operations of analysis, comparison, generalization, and systematization.
Educational: to form the ideological views of students, to instill a sense of success from responsibility for the results obtained.
Lesson type: learning new material.
Equipment: computer, multimedia board.
Expected learning outcomes: the student must
derivative definition
the antiderivative is defined ambiguously.
find antiderivative functions in the simplest cases
check whether the function is antiderivative on a given time interval.
During the classes
Organizing time SLIDE 2
Checking homework
Communicating the topic, purpose of the lesson, objectives and motivation for learning activities.
On the board:
Derivative - produces a new function.
Antiderivative - “primary image”.
4. Updating knowledge, systematizing knowledge in comparison.
Differentiation - finding the derivative.
Integration - restoration of a function from a given derivative.
Introducing new symbols:
5.Oral exercises:SLIDE 3
Instead of points, put some function that satisfies equality.
Students perform self-tests.
adjusting students' knowledge.
5. Studying new material.
A) Reciprocal operations in mathematics.
Teacher: in mathematics there are 2 mutually inverse operations in mathematics. Let's look at it in comparison. SLIDE 4
B) Reciprocal operations in physics.
Two mutually inverse problems are considered in the mechanics section.
Finding the velocity using a given equation of motion of a material point (finding the derivative of a function) and finding the equation of the trajectory of motion using a known velocity formula.
C) The definition of an antiderivative and an indefinite integral is introduced
SLIDE 5, 6
Teacher: in order for the task to become more specific, we need to fix the initial situation.
D) Table of antiderivatives SLIDE 7
Tasks to develop the ability to find antiderivatives - work in groups SLIDE 8
Tasks to develop the ability to prove that an antiderivative is for a function on a given interval - pair work.
6. Physical exerciseSLIDE 9
7. Primary comprehension and application of what has been learned.SLIDE 10
8. Setting homeworkSLIDE 11
9. Summing up the lesson.SLIDE 12
During the frontal survey, together with the students, the results of the lesson are summed up, the concept of new material is consciously comprehended, in the form of emoticons.
I understood everything, managed to do everything.
I didn’t understand part of it, I didn’t manage everything.
Class: 11
Presentation for the lesson
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Technological map of algebra lesson 11th grade.
“A person can recognize his abilities only by trying to apply them.”
Seneca the Younger.
Number of hours per section: 10 hours.
Block topic: Antiderivative and indefinite integral.
Leading topic of the lesson: formation of knowledge and general educational skills through a system of standard, approximate and multi-level tasks.
Lesson objectives:
- Educational: form and consolidate the concept of an antiderivative, find antiderivative functions of different levels.
- Developmental: develop the mental activity of students based on the operations of analysis, comparison, generalization, and systematization.
- Educational: to form the ideological views of students, to instill a sense of success from responsibility for the results obtained.
Lesson type: learning new material.
Teaching methods: verbal, verbal - visual, problematic, heuristic.
Forms of training: individual, pair, group, whole-class.
Means of education: informational, computer, epigraph, handouts.
Expected learning outcomes: the student must
- derivative definition
- the antiderivative is defined ambiguously.
- find antiderivative functions in the simplest cases
- check whether the function is antiderivative on a given time interval.
LESSON STRUCTURE:
- Setting a lesson goal (2 min)
- Preparing to study new materials (3 min)
- Introduction to new material (25 min)
- Initial understanding and application of what has been learned (10 min)
- Setting homework (2 min)
- Summing up the lesson (3 min)
- Reserve jobs.
During the classes
1. Reporting the topic, purpose of the lesson, objectives and motivation for learning activities.
On the board:
***Derivative – “produces” a new function. Antiderivative - primary image.
2. Updating knowledge, systematizing knowledge in comparison.
Differentiation - finding the derivative.
Integration - restoration of a function from a given derivative.
Introducing new symbols:
* oral exercises: instead of dots, put some function that satisfies equality. (see presentation) - individual work.
(at this time, 1 student writes differentiation formulas on the board, 2 students write differentiation rules).
- Self-test is carried out by students. (individual work)
- adjusting students' knowledge.
3. Studying new material.
A) Reciprocal operations in mathematics.
Teacher: in mathematics there are 2 mutually inverse operations in mathematics. Let's look at it in comparison.
B) Reciprocal operations in physics.
Two mutually inverse problems are considered in the mechanics section. Finding the velocity using a given equation of motion of a material point (finding the derivative of a function) and finding the equation of the trajectory of motion using a known velocity formula.
Example 1 page 140 – work with a textbook (individual work).
The process of finding a 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) The definition of an antiderivative is introduced.
Teacher: in order for the task to become more specific, we need to fix the initial situation.
Tasks to develop the ability to find antiderivatives - work in groups. (see presentation)
Tasks to develop the ability to prove that an 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 the error” - individual work. (see presentation)
***perform mutual verification.
Conclusion: when performing these tasks, it is easy to notice that the antiderivative is defined ambiguously.
5. Setting homework
Read the explanatory text chapter 4 paragraph 20, memorize the definition of 1. antiderivative, solve No. 20.1 -20.5 (c, d) - compulsory task for everyone No. 20.6 (b), 20.7 (c, d), 20.8 (b), 20.9 ( b) - 4 examples to choose from.
6. Summing up the lesson.
During the frontal survey, together with the students, the results of the lesson are summed up, the concept of new material is consciously comprehended, in the form of emoticons.
I understood everything, managed to do everything.
I didn’t understand partly, I didn’t manage everything.
7. Reserve tasks.
In case of early completion of the tasks proposed above by the entire class, it is also planned to use tasks No. 20.6(a), 20.7(a), 20.9(a) to ensure employment and development of the most prepared students.
Literature:
- A.G. Mordkovich, P.V. Semenov, Algebra of Analysis, profile level, part 1, part 2 problem book, Manvelov S. G. “Fundamentals of creative lesson development.”
Lesson topic : Antiderivative. Indefinite integral and its properties
Lesson objectives:
Educational:
To introduce students to the concepts of antiderivative and indefinite integral, the main property of antiderivative and the rules for finding antiderivative and indefinite integral.
Educational:
develop independent activity skills,
activate mental activity and mathematical speech.
Educational:
cultivate a sense of responsibility for the quality and results of the work performed;
create responsibility for the final result.
Type lesson : messages of new knowledge
Method of implementation : verbal, visual, independent work.
Security lesson :
Multimedia equipment and software for displaying presentations and videos;
Handout: table of simple integrals (at the consolidation stage).
Lesson structure.
1. Organizational moment (2 min.)
Motivation for 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.)
Progress of the lesson.
Organizing time. (2 minutes.)
Teaching Techniques
Teaching techniques
The teacher greets the students and checks those present in the audience.
Students are preparing for work. The headman fills out a report. The attendants hand out handouts.
Motivation for learning activities.( 5 minutes.)
Teaching Techniques
Teaching techniques
Topic of today's lesson“Primeval.The indefinite integral and its properties.”(Slide 1)
We will use knowledge on this topic in the following lessons when finding certain integrals and areas of plane figures. Much attention is paid to integral calculus in sections of higher mathematics in higher educational institutions when solving applied problems.
Our lesson today is a study of new material, so it will be theoretical in nature. The purpose of the lesson is to form ideas about integral calculus, understand its essence, and 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 Techniques
Teaching techniques
1. We recently covered the topic “Derivatives of some elementary functions.” For example:
Derivative of a functionf (x)= X 9 , We know thatf ′(x)= 9x 8 . Now we will look at an example of finding a function whose derivative is known.
Let's say the derivative is givenf ′(x)= 6x 5 . Using knowledge about the derivative, we can determine that this is the derivative of the functionf (x)= X 6 . A function that can be determined by its derivative is called an antiderivative. (Give a definition of an antiderivative. (slide 3))
Definition 1 : Function F ( x ) is called the antiderivative of the function f ( x ) on the segment [ a; b], if the equality is satisfied 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 an 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 an antiderivative of the functionf (x)= .
Prove with students on the board.
We know that finding the derivative is calleddifferentiation . Finding a function from 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 Fthere is an antiderivative for the functionf, A G– antiderivative forg, That F+ G- there is an antiderivative forf+ g.
(F(x) + G(x))’ = F’(x) + G’(x) = f + g
Rule #2: If F– antiderivative forf, A kis a constant, then the functionkF– antiderivative forkf.
(kF)’ = kF’ = kf
Rule #3: If F– antiderivative forf, A k And b– constants (), then the function
Antiderivative forf(kx+ b).
The history of the concept of integral is closely connected with problems of finding quadratures. Mathematicians of Ancient Greece and Rome called problems about the quadrature of a particular plane figure problems that we now classify as problems for calculating areas. Many significant achievements of mathematicians of Ancient Greece in solving such problems are associated with the use of the exhaustion method proposed by Eudoxus of Cnidus. Using 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 Eudoxus method was improved by Archimedes and the following things were proven:
1. Derivation of the formula for the area of a circle.
2. The volume of the ball is equal to 2/3 of the volume of the cylinder.
All achievements were proven by great mathematicians using integrals.
Let's 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 represents the set of all antiderivative functions forf(x) .
In equality (1) the functionf(x) is called integrand function , and the expression f(x) dx– integrand , variable x – integration variable , term C - integration constant .
Integration is the inverse operation of differentiation. In order to check whether the integration was performed correctly, it is enough to differentiate the result and obtain the integrand function.
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, That
Students record the lecture using handouts and explanations from the teacher. When proving the properties of antiderivatives and integrals, knowledge on the topic of differentiation is used.
4. Table of simple integrals
1. ,( n -1) 2.
3. 4.
5. 6.
The integrals contained in this table are usually calledtabular . Let us note a special case of formula 1:
Let's give another obvious formula:
Algebra lesson in 12th grade.
Lesson topic: “Primordial. Integral"
Goals:
educational
Summarize and consolidate the material on this topic: definition and properties of an antiderivative, table of antiderivatives, rules for finding antiderivatives, the concept of an integral, Newton-Leibniz formula, calculation of areas of figures. To diagnose the assimilation of a system of knowledge and skills and its application to perform practical tasks at a standard level with a transition to a higher level, to promote the development of the ability to analyze, compare, and draw conclusions.
Developmental
perform tasks of increased complexity, develop general learning skills and teach thinking and control and self-control
Educating
Foster a positive attitude towards learning and mathematics
Lesson type: Generalization and systematization of knowledge
Forms of work: group, individual, differentiated
Equipment: cards for independent work, for differentiated work, self-control sheet, projector.
During the classes
Organizing time
Goals and objectives of the lesson: Summarize and consolidate the material on the topic “Antiform. Integral" - definition and properties of an antiderivative, table of antiderivatives, rules for finding antiderivatives, concept of an integral, Newton-Leibniz formula, calculation of areas of figures. To diagnose the assimilation of a system of knowledge and skills and its application to perform practical tasks at a standard level with a transition to a higher level, to promote the development of the ability to analyze, compare, and draw conclusions.
We will conduct the lesson in the form of a game.
Rules:
The lesson consists of 6 stages. Each stage is scored with a certain number of points. On the evaluation sheet you give points for your work at all stages.
Stage 1. Theoretical. Mathematical dictation “Tic Tac Toe”.
Stage 2. Practical. Independent work. Find the set of all antiderivatives.
Stage 3. “Intelligence is good, but 2 is better.” Work in notebooks and 2 students on the board flaps. Find the antiderivative of the function whose graph passes through point A).
4.stage. "Correct mistakes".
5. stage. “Make a word” Calculation of integrals.
6. stage. "Hurry to see." Calculation of the areas of figures bounded by lines.
2. Score sheet.
Mathematicaldictation
Independent work
Verbal response
Correct mistakes
Make up a word
Hurry up to see
9 points
5+1 points
1 point
5 points
5 points
20 points
3 min.
5 minutes.
5 minutes.
6 min
2. Updating knowledge:
stage. Theoretical. Mathematical dictation “Tic Tac Toe”
If the statement is true - X, if false - 0
Function F(x) is called an antiderivative on a given interval if for all x from this interval the equality
The antiderivative of a power function is always a power function
Antiderivative of a complex function
This is the Newton-Leibniz formula
Area of a curved trapezoid
Antiderivative of the sum of functions = the sum of antiderivatives considered on a given interval
Graphs of antiderivative functions are obtained by parallel translation along the X axis to the constant C.
The product of a number and a function is equal to the product of this number and the antiderivative of the given function.
The set of all antiderivatives has the form
Total 9 points
3. Consolidation and generalization
2 stage . Independent work.
“Examples teach better than theory.”
Isaac Newton
Find the set of all antiderivatives:
1 option
The set of all antiderivatives The set of all antiderivativesoption
Self-test.
For correctly completed tasks
Option 1 -5 points,
for option 2 +1 point
1 point for addition.
stage . “The mind is good, and - 2 is better.”
Work on the flaps of the board of two students and all the rest in notebooks.
Exercise
Option 1. Find the antiderivative of the function, the graph of which passes through the point A(3;2)
Option 2. Find the antiderivative of a function whose graph passes through the origin.
Peer review.
For a correct solution -5 points.
stage . Believe it or not, check it if you want.
Task: correct mistakes if they are made.
Find exercises with errors:
Stage . Make up a word.
Evaluate integrals
Option 1.
option.
Answer: BRAVO
Self-test. For a correctly completed task - 5 points.
stage. "Hurry to see."
Calculation areas of figures bounded by lines.
Task: construct a figure and calculate its area.
2 points
2 points
4 points
6 points
6 points
Check individually with the teacher.
For all tasks completed correctly - 20 points
Summarizing:
The lesson covers the main issues
OPEN LESSON ON THE TOPIC
« ANIMID AND INDETERMINATE INTEGRAL.
PROPERTIES OF AN INDETERMINED INTEGRAL".
2 hours.
11th grade with in-depth study of mathematics
Problem presentation.
Problem-based learning technologies.
ANIMID AND INDETERMINATE INTEGRAL.
PROPERTIES OF AN INDETERMINED INTEGRAL.
THE PURPOSE OF THE LESSON:
Activate mental activity;
To promote the assimilation of research methods
- ensure a more durable 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;
develop skills in using the properties of an indefinite integral.
PRELIMINARY WORK:
repeat the rules and formulas of differentiation
concept of differential.
It is proposed to solve problems. The conditions of the tasks are written on the board.
Students give answers to solve problems 1, 2.
(Updating experience in solving problems using differential
citation).
1. Law of body motion S(t), find its instantaneous
speed at any 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 at any
moment of time t.
- I (t) = 6t - 2.
3. Knowing the speed of a moving body at every moment of time,
me, 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 No. 3 and 4 using
the means we have?
(Creating a problematic situation).
Students' assumptions:
- To solve this problem it is necessary to introduce an operation,
the inverse of differentiation.
The differentiation operation compares 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 branch 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 branch of mathematical analysis, together with differential calculus, it forms the basis of the apparatus of mathematical analysis.
Integral calculus arose from the consideration of a large number of problems in natural science and mathematics. The most important of them are the physical problem of determining the distance traveled in a given time using a known, but perhaps variable, speed of movement, and a much more ancient task - calculating the areas and volumes of geometric figures.
What is the uncertainty of this reverse operation remains to be seen.
Let's introduce a definition. (briefly symbolically written
On the desk).
Definition 1. Function F (x) defined on some interval
ke X is called the antiderivative for the given function
on the same interval if for all x 
X
equality holds
F(x) = f (x) or d F(x) = f (x) dx .
For example. (x) = 2x, from this equality it follows that the function
x is antiderivative on the entire number axis
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
noi for the function f if
1) F (x) =
2 cos 2x, f(x) = x 
- 4 sin 2x . 2) F (x) = tan 
x - cos 5x, f(x) = 
+ 5 sin 5x.
3) F (x) = x 
sin x + 
, f (x) = 4x sinx + x cosx + 
.
Students write down the solutions to the examples on the board and comment on them.
ruining your actions.
Is the function x the only antiderivative
for function 2x?
Students give examples
x + 3; x - 92, etc. ,
The students draw their own conclusions:
any function has infinitely many antiderivatives.
Any function of the form x + C, where C is a certain number,
is the antiderivative of the function x.
The antiderivative theorem is written in a notebook under dictation.
teachers.
Theorem. If a function f has an antiderivative on the interval
numeric F, then for any number C the function F + C is also
is an antiderivative of f. Other prototypes
function f on X does not.
The proof is carried out by students under the guidance of a teacher.
a) Because F is an 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 of f on X.
b) Let us prove that the function f of other antiderivatives on X
does not have.
Let us assume that Φ is also antiderivative for f on X.
Then Ф(x) = f(x) and therefore for all x X we have:
F (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, which means any antiderivative
function f on X has the form F + C.
Teacher: what is the task of finding all the prototypes?
nykh for this function?
The students formulate the conclusion:
The problem of finding all antiderivatives is solved
by finding any one: if such a primitive
different is found, then any other is obtained from it
by adding a constant.
The teacher formulates the definition of an indefinite integral.
Definition 2. The set of all antiderivatives of the function f
called the indefinite integral of this
functions.
Designation. 
; 
- read the integral.
= F (x) + C, where F is one of the antiderivatives
for f, C runs through the set
real numbers.
f - integrand function;
f (x)dx - integrand;
x is the integration variable;
C is the constant of integration.
Students study the properties of the indefinite integral independently from the textbook and write them down in their notebooks.
.
Students write down solutions in notebooks, working at the blackboard
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