Antiderivative and integral. Indefinite integral, its properties and calculation. Antiderivative and indefinite integral Concept of antiderivative and indefinite integral presentation

Slide 1

Slide 2

Historical information Integral calculus arose from the need to create a general method for finding areas, volumes and centers of gravity. In its embryonic form, this method was used by Archimedes. He received systematic development in the 17th century in the works of Cavalieri, Torricelli, Fermam, Pascal. In 1659 I. Barrow established a connection between the problem of finding an area and the problem of finding a tangent. Newton and Leib-Nitz in the 70s of the 17th century diverted this connection from the above-mentioned particular geometric problems. At the same time, a connection was established between integral and differential calculus. This connection was used by Newton, Leibniz and their students to develop the technique of integration. Integration methods have reached their current state in the works of L. Euler. The works of MV Ostrogradsko-Go and PL Chebyshev completed the development of these methods.

Slide 3

Integral concept. Let the line MN be given by the equation And it is necessary to find the area F "of the curvilinear trapezoid aABb. We divide the segment ab into n parts (equal or unequal) and construct a stepped figure shown by shading in Fig. 1 Its area, its area is (1) If we enter the designations Then formula (1) will take the form (3) The sought area is the limit of the sum ( 3) for infinitely large n. Leibniz introduced the notation for this limit (4) In which (italic s) is the initial letter of the word summa (sum), the E expression indicates the typical form of individual terms - Ms. Leibniz began to call the expression integral - from the Latin word integralis - integral. JB Fourier improved Leibniz's notation, giving it the form Here, the initial and final values \u200b\u200bof x are explicitly indicated.

Slide 4

The relationship between integration and differentiation. We will consider a constant, and b - variable. Then the integral will be a function of b. The differential of this function is

Slide 5

Antiderivative function. Let the function be the derivative of the function, T.S. There is a differential of the function: Then the function is called the antiderivative for the function

Slide 6

An example of finding the antiderivative. The function is the antiderivative of TS. There is a differential of a function The function is the antiderivative of a function

Slide 7

Indefinite integral. The most general form of its antiderivative function is called the indefinite integral of this expression. The indefinite integral of the expression is denoted The expression is called the integrand, the function is the integrand, the variable x is the variable of integration. Finding the indefinite integral of a given Function is called integration. Anoshina O.V.

Main literature

1. Shipachev V. S. Higher mathematics. Basic course: textbook and
workshop for bachelors [stamp of the RF Ministry of Education] / V.S.
Shipachev; ed. A.N. Tikhonov. - 8th ed., Rev. and add. Moscow: Yurayt, 2015 .-- 447 p.
2. Shipachev V. S. Higher mathematics. Complete course: tutorial
for acad. bachelor degree [Grif UMO] / VS Shipachev; ed. A.
N. Tikhonova. - 4th ed., Rev. and add. - Moscow: Yurayt, 2015 .-- 608
from
3. Danko P.E., Popov A.G., Kozhevnikova T.Ya. Higher mathematics
in exercises and tasks. [Text] / P.E. Danko, A.G. Popov, T. Ya.
Kozhevnikov. At 2 pm - M .: Higher school, 2007 .-- 304 + 415c.

Reporting

1.
Test. Performed in accordance with:
Tasks and guidelines for the implementation of control works
in the discipline "APPLIED MATHEMATICS", Yekaterinburg, FGAOU
VO "Russian State Professional and Pedagogical
University ", 2016 - 30s.
Select the test option by the last digit of the number
grade book.
2.
Exam

Indefinite integral, its properties and calculation Antiderivative and indefinite integral

Definition. The function F x is called
antiderivative function f x defined on
some interval if F x f x for
each x from this interval.
For example, the function cos x is
the antiderivative of the sin x function, since
cos x sin x.

Obviously, if F x is the antiderivative
function f x, then F x C, where C is some constant, is also
antiderivative function f x.
If F x is any antiderivative
function f x, then any function of the form
Ф x F x C is also
antiderivative function f x and any
the antiderivative is representable in this form.

Definition. The totality of all
antiderivatives of the function f x,
identified at some
interval is called
indefinite integral of
function f x on this interval and
denoted f x dx.

If F x is some antiderivative of the function
f x, then they write f x dx F x C, although
it would be more correct to write f x dx F x C.
According to the established tradition, we will write
f x dx F x C.
Thus, the same symbol
f x dx will denote as all
the set of antiderivatives of the function f x,
and any element of this set.

Integral properties

The derivative of the indefinite integral is
the integrand, and its differential is the subintegral expression. Really:
1. (f (x) dx) (F (x) C) F (x) f (x);
2.d f (x) dx (f (x) dx) dx f (x) dx.

Integral properties

3. Indefinite integral of
differential continuously (x)
of the differentiable function is equal to
this function up to a constant:
d (x) (x) dx (x) C,
since (x) is antiderivative for (x).

Integral properties

4. If the functions f1 x and f 2 x have
antiderivatives, then the function f1 x f 2 x
also has an antiderivative, and
f1 x f 2 x dx f1 x dx f 2 x dx;
5. Kf x dx K f x dx;
6.f x dx f x C;
7.f x x dx F x C.

1.dx x C.
a 1
x
2.x a dx
C, (a 1).
a 1
dx
3.ln x C.
x
x
a
4.a x dx
C.
ln a
5.e x dx e x C.
6.sin xdx cos x C.
7.cos xdx sin x C.
dx
8.2 ctgx C.
sin x
dx
9.2 tgx C.
cos x
dx
arctgx C.
10.
2
1 x

Indefinite Integral Table

11.
dx
arcsin x C.
1 x 2
dx
1
x
12.2 2 arctan C.
a
a
a x
13.
14.
15.
dx
a2 x2
x
arcsin C ..
a
dx
1
x a
ln
C
2
2
2a x a
x a
dx
1
a x
a 2 x 2 2a ln a x C.
dx
16.
x2 a
ln x x 2 a C.
17.shxdx chx C.
18.chxdx shx C.
19.
20.
dx
ch 2 x thx C.
dx
cthx C.
2
sh x

Differential properties

When integrating, it is convenient to use
properties: 1
1.dx d (ax)
a
1
2.dx d (ax b),
a
1 2
3.xdx dx,
2
1 3
2
4.x dx dx.
3

Examples of

Example. Evaluate cos 5xdx.
Decision. In the table of integrals we find
cos xdx sin x C.
We transform this integral to a tabular one,
taking advantage of the fact that d ax adx.
Then:
d 5 x 1
\u003d cos 5 xd 5 x \u003d
cos 5xdx cos 5 x
5
5
1
\u003d sin 5 x C.
5

Examples of

Example. Calculate x
3x x 1 dx.
Decision. Since under the integral sign
the sum of four terms is found, then
expand the integral into the sum of four
integrals:
2
3
2
3
2
3
x
3
x
x
1
dx
x
dx
3
x
dx xdx dx.
x3
x4 x2
3
x C
3
4
2

Variable independence

When calculating integrals, it is convenient
use the following properties
integrals:
If f x dx F x C, then
f x b dx F x b C.
If f x dx F x C, then
1
f ax b dx F ax b C.
a

Example

Let's calculate
1
6
2
3
x
dx
2
3
x
C
.
3 6
5

Integration methods Integration by parts

This method is based on the udv uv vdu formula.
The following integrals are taken by the method of integration by parts:
a) x n sin xdx, where n 1,2 ... k;
b) x n e x dx, where n 1,2 ... k;
c) x n arctgxdx, where n 0, 1, 2, ... k. ;
d) x n ln xdx, where n 0, 1, 2, ... k.
When calculating integrals a) and b), introduce
n 1
notation: x n u, then du nx dx, and, for example
sin xdx dv, then v cos x.
When calculating the integrals c), d) denote by u the function
arctgx, ln x, and for dv take x n dx.

Examples of

Example. Evaluate x cos xdx.
Decision.
u x, du dx
=
x cos xdx
dv cos xdx, v sin x
x sin x sin xdx x sin x cos x C.

Examples of

Example. Calculate
x ln xdx
dx
u ln x, du
x
x2
dv xdx, v
2
x2
x 2 dx
ln x
=
2
2 x
x2
1
x2
1 x2
ln x xdx
ln x
C.
=
2
2
2
2 2

Variable replacement method

Let it be required to find f x dx, and
directly pick the antiderivative
for f x we \u200b\u200bcannot, but we know that
she exists. You can often find
antiderivative, introducing a new variable,
according to the formula
f x dx f t t dt, where x t and t is a new
variable

Integration of functions containing a square trinomial

Consider the integral
ax b
dx,
x px q
containing a square trinomial in
denominator of the integrand
expressions. Such an integral is also taken
variable change method,
pre-highlighting in
the denominator is a full square.
2

Example

Calculate
dx
.
x 4x 5
Decision. Convert x 2 4 x 5,
2
selecting a complete square according to the formula a b 2 a 2 2ab b 2.
Then we get:
x2 4x 5 x2 2 x 2 4 4 5
x 2 2 2 x 4 1 x 2 2 1
x 2 t
dx
dx
dt
x t 2
2
2
2
x 2 1 dx dt
x 4x 5
t 1
arctgt C arctg x 2 C.

Example

To find
1 x
1 x
2
dx
tdt
1 t
2
x t, x t 2,
dx 2tdt
2
t2
1 t
2
dt
1 t
1 t
d (t 2 1)
t
2
1
2
2tdt
2
dt
ln (t 1) 2 dt 2
2
1 t
ln (t 2 1) 2t 2arctgt C
2
ln (x 1) 2 x 2arctg x C.
1 t 2 1
1 t
2
dt

A definite integral, its basic properties. Newton-Leibniz formula. Certain integral applications.

The concept of a definite integral is led by
the problem of finding the area of \u200b\u200ba curvilinear
trapezium.
Let on some interval be given
continuous function y f (x) 0
A task:
Plot its graph and find the F area of \u200b\u200bthe figure,
bounded by this curve, by two straight lines x \u003d a and x
\u003d b, and from below - a segment of the abscissa axis between the points
x \u003d a and x \u003d b.

The figure aABb is called
curved trapezoid

Definition

b
f (x) dx
Under a definite integral
a
of a given continuous function f (x) on
this segment is understood
the corresponding increment of it
antiderivative, that is
F (b) F (a) F (x) /
b
a
The numbers a and b are the limits of integration,
- the interval of integration.

Rule:

The definite integral is equal to the difference
values \u200b\u200bof the antiderivative integrand
functions for upper and lower limits
integration.
Introducing the notation for the difference
b
F (b) F (a) F (x) / a
b
f (x) dx F (b) F (a)
a
Newton-Leibniz formula.

Basic properties of a definite integral.

1) The value of the definite integral does not depend on
the designation of the variable of integration, i.e.
b
b
a
a
f (x) dx f (t) dt
where x and t are any letters.
2) A definite integral with the same
outside
integration is zero
a
f (x) dx F (a) F (a) 0
a

3) When interchanging the limits of integration
definite integral reverses sign
b
a
f (x) dx F (b) F (a) F (a) F (b) f (x) dx
a
b
(additivity property)
4) If the interval is divided into a finite number
partial intervals, then a definite integral,
taken over the interval is equal to the sum of certain
integrals taken over all its partial intervals.
b
c
b
f (x) dx f (x) dx
c
a
a
f (x) dx

5) Constant multiplier can be taken out
for the sign of a definite integral.
6) A definite integral of an algebraic
sums of a finite number of continuous
functions is equal to the same algebraic
the sum of definite integrals of these
functions.

3. Change of variable in a definite integral.

3. Replacing a variable in a specific
integral.
b
f (x) dx f (t) (t) dt
a
a (), b (), (t)
Where
for t [; ], functions (t) and (t) are continuous on;
5
Example:
1
=
x 1dx
=
x 1 5
t 0 4
x 1 t
dt dx
4
0
3
2
t dt t 2
3
4
0
2
2
16
1
t t 40 4 2 0
5
3
3
3
3

Improper integrals.

Improper integrals.
Definition. Let the function f (x) be defined on
infinite interval, where b< + . Если
exists
b
lim
f (x) dx,
b
a
then this limit is called improper
integral of the function f (x) on the interval
}