slide 1
slide 2
Historical information Integral calculus arose from the need to create a general method of Finding areas, volumes and centers of gravity. In its embryonic form, this method was used by Archimedes. It received systematic development in the 17th century in the works of Cavalieri, Torricelli, Fermam, and 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 mentioned particular geometrical problems. Thus, 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 reached their current state mainly in the works of L. Euler. The works of M.V. Ostrogradsko-Go and P.L. Chebyshev completed the development of these methods.
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The concept of the integral. Let the line MN be given by the equation And we need to find the area F of the curvilinear trapezoid aABb. Let us divide the segment ab into n parts (equal or unequal) and construct a stepped figure shown by hatching in Fig. 1 Its area, its area is equal to (1) If we introduce the notation Then, formula (1) will take the form (3) The desired area is the limit of the sum ( 3) for infinitely large n. Leibniz introduced the designation 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. Leibniz began to call the expression integral - from the Latin word integralis - integral. J. B. Fourier improved Leibniz's notation, giving it the form Here, the initial and final values of x are explicitly indicated.
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Relationship between integration and differentiation. Let us consider a constant and b variable. Then the integral will be a function of b . The differential of this function is
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primitive function. Let a function be a derivative of a function, T.S. There is a function differential: Then the function is called antiderivative for the function
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An example of finding an antiderivative. The function is the antiderivative from T.S. There is a differential of the function The function is the antiderivative of the function
Slide 7
Indefinite integral. The indefinite integral of a given expression is the most general form of its antiderivative function. The indefinite integral of an expression is denoted The expression is called a subintegral expression, the function is called a subintegral function, the variable x is the variable of integration. Finding the indefinite integral of a given Function is called integration.
GBOU SPO "Navashinsky Ship Mechanical College" Indefinite integral. Calculation methods
Eudoxus of Knidos c. 408 - ca. 355 BC e. 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
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 + CF(x) = x ³ + CF(x) = - cos x + 5x + CF(x) = 5 sin x + CF(x) = 2 x ³ + CF(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 andworkshop 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
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 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 calledantiderivative 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 isintegrand, 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 ofdifferential 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 haveantiderivatives, 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 useproperties: 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 x3x 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 convenientuse 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
Compute1
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. Calculatex 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 , anddirectly 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 integralaxb
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
Calculatedx
.
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
To find1 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 tothe 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
bf(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 differencevalues 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 onintegration 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 certainintegral.
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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