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Файл:Mathematics for Foreign Students integrals. Study Guide
.pdf
The Ministry of Science and Higher Education of the Russian Federation
Kazan National Research Technological University
MATHEMATICS FOR FOREIGN
STUDENTS: INTEGRALS
Study Guide
Kazan
KNRTU Press
2022
UDC 51(075)
Published by the decision of the Editorial Review Board
D. Bikmukhametova
Mathematics for Foreign Students: Integrals : Study Guide / D. Bikmukhametova, N. Gazizova, S. Enikeeva [et al.]; The Ministry of Education
and Science of the Russian Federation, Kazan National Research Technological University. – Kazan: KNRTU Press, 2022. – 100 p.
ISBN 978-5-7882-3290-4
ISBN 978-5-7882-3290-4
©
D. Bikmukhametova, N. Gazizova,
S. Enikeeva, A. Mindubaeva, N. Nikonova,
S. Alkhaleefah, 2022
©
Kazan National Research Technological
University, 2022
UDK UDC 51(075)
of the Kazan National Research Technological University
Reviewers:
PhD in Physics and Mathematics, Associate Professor A. Antonova
PhD in Physics and Mathematics, Full Professor I. Kayumov
The Study Guide contains and encompasses theoretical information on
the applied problems of how to introduce a calculus or differential calculus.
It is intended for students studying higher Mathematics and Calculus.
This Study Guide is prepared at the Department of Advanced Mathematics.
2
C O N T E N T S
INTRODUCTION ............................................................................................ 4
INDEFINITE INTEGRALS ................................................................................. 5
INTEGRATION OF TRIGONOMETRICAL FUNCTIONS ................................... 27
THE UNIVERSAL TRIGONOMETRIC SUBSTITUTION ..................................... 32
DEFINITE INTEGRAL, METHODS OF ITS CALCULATION ............................... 46
INTEGRATION OF RATIONAL FRACTIONS ................................................... 53
APPLICATIONS OF DEFINITE INTEGRALS ..................................................... 61
REFERENCES ................................................................................................ 97
3
I N T R O D U C T I O N
This Study Guide is intended for foreign and even for international 1st
and 2nd-year full-time, internal, and part-time students studying higher
Mathematics and Calculus. It fully covers the material in Advanced Mathematics for students majoring in engineering.
Studying this Guide allows forming the general cultural and the professional competencies for students:
– Independent work skills;
– Self-organization and self-education skills;
– Skills to generalize and analyze information, set goals and select
ways to achieve goals;
– Readiness to apply fundamental mathematical, natural science, and
general engineering knowledge in general professional activities.
To prepare the present Study Guide we considered, studied and analyzed the works of Russian and foreign authors. The Study Guide consists of
the theoretical part, examples about solving typical problems, tests, and answer keys. The theoretical part includes all necessary information on how to
prepare for tests, colloquia and final examination. Texts are illustrated with
a large number of examples and figures.
In addition to the basic formulas and definitions, the authors offer a detailed analysis of the tests. The Study Guide provides a set of tasks that can
be used by both lecturers for organizing their classes and classroom tests and
students for their self-study and self-preparation for tests, colloquia and examinations.
4
I N D E F I N I T E I N T E G R A L S
( )
xx
dx
d
cossin =
+= Cxxdx sincos
k
1
Cxdx +=
C
n
x
dxx
n
n
+
+
=
+
1
1
The Definition and Properties
In general we can say that the integration is the reverse of differentia-
tion so that we have the following definition:
Definition: The function F(x) is called a primitive (or antiderivative)
of a function f (x), if the equality F′(x) = f (x) is hold for all x into the domain
of f (x), and the set of all primitives F(x) of f (x) is called the indefinite integral
of the function f(x). The indefinite integral of f(x) is denoted by the symbol
f(x)dx=F(x)+С.
The process of finding antiderivatives is called antidifferentiation,
more commonly referred to it as integration.
Standard integrals
For every differential coefficient, when it’s written in reverse, gives
us standard integral,
Properties of Indefinite Integrals
Property 1. (f(x)dx) =f(x), df(x)dx=f(x)dx.
Property 2. dF(x)=F(x) + С, dx=x+С.
Property 3. (f1(x) f2(x))dx = f1(x)dx f2(x)dx
Property 4. cf(x)dx=cf(x)dx, c=const.
Property 5. ∫f(kx+b)dx=
F(kx+b)+C.
A Table of Common Integrals
1)
; 2)
, n –1;
5
6
3)
Cx
x
dx
+=
||ln
; 4)
Cedxe
xx
+=
;
5)
C
a
a
dxa
x
x
+=
ln
(a>0, a≠1); 6)
Cxxdx +−=
cossin
;
7)
Cxxdx +=
sincos
; 8)
Cxtgxdx +−=
|cos|ln
;
9)
Cxctgxdx +=
|sin|ln
; 10)
Ctgx
x
dx
+=
2
cos
;
11)
Cctgx
x
dx
+−=
2
sin
; 12)
+C
a
x
arctg
axa
dx 1
22
=
+
;
13)
C
ax
ax
a
ax
dx
+
+
−
=
−
ln
2
1
22
; 14)
C
a
x
xa
dx
+=
−
arcsin
22
;
15)
Caxx
ax
dx
++=
22
22
ln
;
16)
−
2
1 x
dx
=arcsinx + C= – arccosx +C;
17)
+1
2
x
dx
=arctgx + C = –arcctgx +C.
Example. To solve
1.
( )
++ dxxxx cos52
23
.
Solution: From the properties of integrals we have
( )
++ dxxxx cos52
23
=
=
++ xdxdxxdxx cos52
23
;sin5
3
2
4
34
Cxxx+++=
Another examples. Find the following integrals
2.
( )
CxxxCx
xx
dxxx +++=+++=++
727
4
4
3
3743
23
23
2
;
7
3.
=
++=
++
−
dxxx
x
dxx
x
x
3
2
3
3
2
3
86
7
8
67
=++−=++
−
+=
−
−
CxxxC
xx
x
3
5
2
3
5
2
5
24
3ln7
3
5
8
2
6ln7
Cx
x
x ++−=
3
5
2
5
243
ln7
;
4.
=
++=
++
−
dxxxdxx
x
xx
cos329cos32
9
2
3
3
Cx
x
x
+++
−
=
−
sin3
2ln
2
2
1
9
2
1
Cx
x
x
+++−= sin3
2ln
218
;
5.
+−=
+− Cxxdxtgxx cosln5cos
3
2
5sin
3
2
;
6.
( )
++=
+
+
−
Carctgxxdx
x
x
5
6
arcsin4
15
6
1
4
2
2
;
7.
+
+
−
=
−
C
x
x
x
dx
7
7
ln
7
4
49
8
2
;
8.
++=
+
+=
+
++
=
+
+
Carctgxxdx
x
dx
x
x
dx
x
x
4
1
4
1
1
41
1
5
22
2
2
2
;
9.
( )
.
4
1
2
−xx
We can rewrite this integral by mathematical formu-
las and so we get that,
8
( )
( )
=+−=
+−
=
−
−
dxxxxdx
x
xx
x
x
4/14/14/3
4/14
2
2
121
.
3
4
5
4
2
7
4
4/34/54/7
C
xxx
++−=
10.
=
−
+−−
dx
x
xx
9
33
4
22
( )( ) ( )( )
=
+−
+
−
+−
−
=
dx
xx
x
xx
x
33
3
33
3
22
2
22
2
=
−
−
+
= dx
xx 3
1
3
1
22
Cxxxx +−−−++ 3ln3ln
22
;
11.
+
=
=
=
C
e
e
dx
e
dx
e
dx
e
x
xx
x
xx
10
ln
10
105252
;
12.
Cxdxxdx
x
x
dx
x
x
+−===
−
cos2sin2
sin
sin2
sin
2cos1
2
;
13.
C
x
C
x
x
dx
x
dx
+=+=
−
=
−
2
3
arcsin
3
1
3
2
arcsin
3
1
9
4
3
1
94
2
2
.
Do the following exercises and test Yourself
1.
dx4
; 2.
dtt
2
3
;
3.
−
dxx32
; 4.
ds
;
9
5.
dxx
4
3
21
; 6.
dxx
5
12
;
7.
3
2x
dx
; 8.
2
4u
du
;
9.
5
3
x
dx
; 10.
( )
+ dxx 64
3
;
11.
( )
−+ dxxx 265
24
; 12.
−+ dxxx 14
3
4
;
13.
+
t
dtt )4(
; 14.
du
u
u
+12
2
;
15.
dz
z
z
+
2
3
; 16.
( )
+ dxxx 43
;
17.
( )
+ duuu 24
2
; 18.
( )
−− dxxx 56)1(
;
19.
dxxx
2
7
; 20.
− dxctgx
x
4
cos
4
2
;
21.
++− dxe
x
xx
23
sin2
5
2
; 22.
−
−
+
dx
xx 9
11
5
3
22
;
23.
+ 49
4
2
x
dx
; 24.
( )
−
2
162
5
x
dx
;
25.
+−
dx
xx
xx 43
2
; 26.
−237 x
dx
;
10
27.
dx
x
xx
+−
2
431
;
28.
+−
dx
x
xxx 4713
4
5
4
.
Answers:
1)
Cx +4
; 2)
Ct +
3
;
3)
Cx +−
−2
; 4)
Cs+
;
5)
Cx +
4
7
12
; 6)
Cxx +
5
10
;
7)
C
x
+−
2
4
1
; 8)
C
u
+−
4
1
;
9)
C
x
+
2
5
5
2
; 10)
Cxx ++ 6
4
;
11)
Cxxx +−+ 22
35
; 12)
Cxx
x
+−+
2
3
7
2
7
3
;
13)
Ctt ++ ||ln4
; 14)
Cuu ++ ||ln
2
;
15)
C
z
z +−
3
||ln
; 16)
Cxx ++
23
2
;
17)
Cuu ++
24
; 18)
Cxxx ++− 5
2
11
2
23
;
19)
Cxx +
3
2
; 20)
Cxtgx +− sinln44
;
21)
Cectgx
x
x
+++ 2
3ln
3
2
5
;
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