INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 03,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 828
METHODS FOR SOLVING DEFINITE INTEGRALS
Sadiqova Ixbol Iskanderovna
Jizzakh state pedagogy university ,
Mathematician teaching methodology department teacher
e-mail:
Abstract:
This article unclear integrals solution two main to the method , that is in pieces
integration ( partial integration ) and variable exchange to the methods dedicated . This methods ,
unclear integrals in calculation complicated functions simplification and to calculate facilitate for
wide is applied .The article firstly breaks down of integration main principles and his/her unclear
integrals in solution how application seeing We will go out . This The method usually involves
integrals simpler to expressions separation through the solution to find help gives .
Key words :
Indefinite integral, piecewise integration , variable exchange , mathematics analysis ,
integrals solve , functions simplification , variable exchange
In our country, mathematics has been identified as one of the priority areas for the development
of science in 2020. Over the past period, a number of systematic works have been carried out
aimed at bringing mathematical science and education to a new qualitative stage.
Resolution of the President of the Republic of Uzbekistan dated May 7, 2020 No. PQ-4708 “On
measures to improve the quality of education and develop scientific research in the field of
mathematics”, dated July 9, 2020 “On state support for the further development of mathematical
education and sciences, as well as the V.I. In connection with the rapid development of modern
branches of mathematics, science and technology, the foundation of which was laid by our great
ancestors such as Muhammad Al-Khwarizmi, Ahmad Al-Farghani, Abu Rayhon Beruni, Mirzo
Ulugbek, and the resolutions of the State Duma No. PQ-4387 “On measures to radically improve
the activities of the Romanovsky Institute of Mathematics”, the in-depth teaching of integrals,
one of the main topics in higher education, is currently one of the important issues. This article
serves to address the above-mentioned resolutions and the issues set before us.
Taking the above into account, we will study several methods for finding definite integrals of the
following form.
If
the equality holds
for ,
the function is called the
initial function of a
continuous
function on the interval.
Given function elementary function to find integration The function is called of integration one
how many methods there is is , this in the article in pieces integration method and variables
replacement method about stopped Let's go .
1. Integration by parts method .
If u(x) and v(x) are differentiable functions if so , in
pieces integration for
Formula (1) is used. This method
is
used
to
find
integrals
of
the
form
,
,
,
,
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 03,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 829
,
,
, .
This method in use under the given integral f(x)dx
expression he *dv in appearance
choice must be , as a result of (1) – right on the side voodoo expression in integration initial
integral to find relatively simple Let it be .
In general he and dv what to choose separately attention to give necessary.If given integral
under expression plural and trigonometric or indicative functions from the multiplication
consists of if , then she is function as polytheism to take to the goal appropriate will be .
If the integral is function logarithmic and reverse trigonometric of functions in the plural
consists of if , then she is function as from differentiation then soda to look
coming function
if taken good result gives .
In pieces integration repetitive in use initially she is with how function what designated if we
are then he /she with this function designation necessary , otherwise integral to zero without
equal will be .
Example 1.
Find the integral.
Solution
:
u=3x+5
and
We
get
uv
=
cos5x
.
Then
du=3dx,
Substituting this expression into formula (1), we obtain the
following:
Example 2.
Find the integral.
Solution :
(*)
Harvest made (*) PCB right by
To find the integral, we again use the formula for
integration by parts:
xln3xdx =
u = ln3x dv = xdx
du =
1
x
dx
v =
x
2
2
=
x
2
2 ln3x −
x
2
2 ln3xdx ==
x
2
2 ln3x −
x
2
4 + C
This is related to (*) given , given function integral
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 03,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 830
w
e determine that.
2. Substitution of variables method .
Hypothesis let's do it Let the substitution be performed
in the integral
. If
a continuous function with an inverse has a
derivative
, then
(2)
The formula is valid. will be . (2) to unclear in the integral variable replacement formula It is
called .
Some in cases
Getting a replacement will yield good results.
The variable replacement given internal simple to look if it brings it , application to the
goal appropriate will be .
Uncertain in the integral variable replacement as a result found elementary in function
new variable instead of initial variable through expression leaving given of the integral answer
is defined . The function differential inside input , variable replacement method private is the
case .
Example 3.
Find the integral.
Solution
:
by
equality
variable
we
replace
,
then
and
Example 4.
Find the integral .
Solution :
We substitute the variable through the equation, then
dt = 4cos4xdx <=> cos4xdx =
1
4 dt
sin4x + 3cos4xdx =
1
4
t
1
2
dt =
1
4
t
3
2
3
2
+ C =
1
6
t
3
+ C
t=sin4x+3
=
1
6
sin4x + 3
3
+ C
Done replacement
differential of a function under to enter equivalent .
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 03,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 831
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Berman , Sbornik zadach po kursu matematicheskogo analiza.- M. , "Nauka"
1971 g.
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1974.
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