Methods of calculation". Presentation for the lesson "Indefinite integral. Calculation methods "Extrema of a function of two variables

GBOU SPO "Navashinsky Ship Mechanical College" Indefinite integral. Calculation methods

Eudoxus of Knidos c. 408 - ca. 355 BC e. The integral calculus appeared during the ancient period of the development of mathematical science and began with the method of exhaustion, which was developed by the mathematicians of ancient Greece, and was a set of rules developed by Eudoxus of Cnidus. According to these rules, the areas and volumes were calculated

Leibniz Gottfried Wilhelm (1646-1716) The symbol ∫ was introduced by Leibniz (1675). This sign is a variation of the Latin letter S (the first letter of the word summa).

Gottfried Wilhelm Leibniz (1646-1716) Isaac Newton (1643 - 1727) Newton and Leibniz independently discovered a fact known as the Newton-Leibniz formula.

Augustin Louis Cauchy (1789 - 1857) Carl Theodor Wilhelm Weierstrass (1815 1897) The work of Cauchy and Weierstrass summed up the centuries-old development of integral calculus.

Russian mathematicians took part in the development of integral calculus: M.V. Ostrogradsky (1801 - 1862) V.Ya. Bunyakovsky (1804 - 1889) P.L. Chebyshev (1821 - 1894)

UNDEFINITE INTEGRAL An indefinite integral of a continuous function f(x) on the interval (a; b) is any of its antiderivative functions. Where C is an arbitrary constant (const).

1. f(x) = x n 2. f(x) = C 3. f(x)= sinx 4. f(x) = 6. f(x) = 1. F(x) = Cx + C 2 F(x) = 3. F(x) = 4. F(x) = sin x + C 5. F(x) = c tg x + C 6. F (x) = - cos x + C 5. f(x) = cosx Match. Find such a general form of the antiderivative that corresponds to the given function. tgx +С

Integral Properties

Basic methods of integration Tabular. 2. Reduction to a tabular transformation of the integrand into a sum or difference. 3.Integration using a change of variable (substitution). 4. Integration by parts.

Find antiderivatives for functions: F(x) = 5 x ² + C F(x) = x ³ + C F(x) = - cos x + 5x + C F(x) = 5 sin x + C F(x) = 2 x ³ + C F(x) = 3 x - x ² + C 1) f(x) = 10x 2) f(x) =3 x ² 3) f(x) = sin x +5 4) f(x) = 5 cos x 5) f (x) \u003d 6 x ² 6) f (x) \u003d 3-2x

Is it true that: a) c) b) d)

Example 1. The integral of the sum of expressions is equal to the sum of the integrals of these expressions. A constant factor can be taken out of the integral sign

Example 2. Check solution Record solution:

Example 3. Check solution Record solution:

Example 4 . Check the solution Write the solution: Introduce a new variable and express the differentials:

Example 5. Check solution Record solution:

C independent work Find the indefinite integral Check the solution Level "A" (by "3") Level "B" (by "4") Level "C" (by "5")

Task Establish a match. Find such a general form of the antiderivative that corresponds to the given function.

Anoshina O.V.

Main literature

1. V. S. Shipachev, Higher Mathematics. Basic course: textbook and
workshop for bachelors [Certificate of the Ministry of Education of the Russian Federation] / V. S.
Shipachev; ed. A. N. Tikhonova. - 8th ed., revised. and additional Moscow: Yurayt, 2015. - 447 p.
2. V. S. Shipachev, Higher Mathematics. Full course: textbook
for acad. Bachelor's degree [Certificate of UMO] / V. S. Shipachev; ed. BUT.
N. Tikhonova. - 4th ed., Rev. and additional - Moscow: Yurayt, 2015. - 608
With
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 o'clock - M .: Higher School, 2007. - 304 + 415c.

Reporting

1.
Test. Performed in accordance with:
Tasks and guidelines for the performance of examinations
in the discipline "APPLIED MATHEMATICS", Yekaterinburg, FGAOU
VO "Russian State Vocational Pedagogical
University", 2016 - 30s.
Choose the option of control work by the last digit of the number
record 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 cos x function is
antiderivative function sin x , since
cos x sin x .

Obviously, if F x is an antiderivative
functions f x , then F x C , where C is some constant, is also
antiderivative function f x .
If F x is some antiderivative
function f x , then any function of the form
F x F x C is also
antiderivative function f x and any
primitive can be represented in this form.

Definition. The totality of all
antiderivatives of the function f x ,
defined on some
in between is called
indefinite integral of
functions f x on this interval and
denoted by 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 .
We, according to the established tradition, will write
f x dx F x C .
Thus the same symbol
f x dx will denote as the whole
set of antiderivatives of the function f x ,
and any element of this set.

Integral Properties

The derivative of the indefinite integral is
integrand, and its differential to the integrand. 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)
differentiable function is equal to itself
this function up to a constant:
d (x) (x) dx (x) C,
since (x) is an antiderivative of (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 Kf 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. 2tgx C .
cos x
dx
arctgx C .
10.
2
1 x

Table of indefinite integrals

11.
dx
arcsin x C .
1x2
dx
1
x
12. 2 2 arctan C .
a
a
a x
13.
14.
15.
dx
a2x2
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 log a x C .
dx
16.
x2 a
log 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

Properties of differentials

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

Examples

Example. Calculate cos 5xdx .
Solution. In the table of integrals we find
cos xdx sin x C .
Let us transform this integral to a tabular one,
taking advantage of the fact that d ax adx .
Then:
d5 x 1
= cos 5 xd 5 x =
cos 5xdx cos 5x
5
5
1
= sin 5 x C .
5

Examples

Example. Calculate x
3x x 1 dx .
Solution. Since under the integral sign
is the sum of four terms, then
expand the integral as a 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

Independence of the type of variable

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

Compute
1
6
2
3
x
dx
2
3
x
C
.
3 6
5

Integration methods Integration by parts

This method is based on the formula udv uv vdu .
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 the integrals a) and b) enter
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 for u the function
arctgx , ln x , and for dv they take x n dx .

Examples

Example. Calculate x cos xdx .
Solution.
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

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
1x2
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 up the primitive
for f x we cannot, but we know that
she exists. Often found
antiderivative by introducing a new variable,
according to the formula
f x dx f t t dt , where x t and t is the new
variable

Integration of functions containing a square trinomial

Consider the integral
axb
dx ,
x px q
containing a square trinomial in
the denominator of the integrand
expressions. Such an integral is also taken
change of variables method,
previously identified in
the denominator is a full square.
2

Example

Calculate
dx
.
x4x5
Solution. Let's transform x 2 4 x 5 ,
2
selecting a full square according to the formula a b 2 a 2 2ab b 2 .
Then we get:
x2 4x5 x2 2x2 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
x4x5
t1
arctgt C arctg x 2 C.

Example

Find
1 x
1 x
2
dx
tdt
1 t
2
x t, x t 2 ,
dx2tdt
2
t2
1 t
2
dt
1 t
1 t
d (t 2 1)
t
2
1
2
2tdt
2
dt
log(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

Definite integral, its main properties. Newton-Leibniz formula. Applications of a definite integral.

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

The figure aABb is called
curvilinear trapezoid

Definition

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

Rule:

The definite integral is equal to the difference
values of 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 a definite integral does not depend on
integration variable notation, 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 rearranging the limits of integration
the definite integral reverses its 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 the definite integral,
taken over the interval is equal to the sum of the defined
integrals taken over all its partial intervals.
b
c
b
f(x)dx f(x)dx
c
a
a
f(x)dx

5) A constant multiplier can be taken out
for the sign of a definite integral.
6) A definite integral of the 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 certain
integral.
b
f (x)dx f (t) (t)dt
a
a(), b(), (t)
where
for t[; ] , the functions (t) and (t) are continuous on;
5
Example:
1
=
x 1dx
=
x 1 5
t04
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
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