Mathematics for Foreign Students: Integrals

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 
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 
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 
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. 
UDK  UDC 51(075) 
 
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 
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
The Definition and Properties 
In general we can say that the integration is the reverse of differentiation 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, 
(
)
x
x
dx
d
cos
sin
=

+
=

C
x
xdx
sin
cos
Properties of Indefinite Integrals 
Property 1. (f(x)dx) =f(x), 
df(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=cf(x)dx, 
c=const.  
Property 5. ∫f(kx+b)dx=
k
1 F(kx+b)+C.
A Table of Common Integrals 
1
 , n –1;  
1)
C
x
dx
+
=

;  
2) 
C
n
x
dx
x
n
n
+
+
=
+

1
5 

dx
+
=

|
|
ln
;  
4) 
C
e
dx
e
x
x
+
=

;  
3) 
C
x
x
a
dx
a
(a>0, a≠1); 
6) 
C
x
xdx
+
−
=

cos
sin
; 
5) 
C
a
x
x
+
=

ln
7) 
C
x
xdx
+
=

sin
cos
;  
8) 
C
x
tgxdx
+
−
=

|
cos
|
ln
; 
; 
9) 
C
x
ctgxdx
+
=

|
sin
|
ln
;  
10) 
C
tgx
x
dx
+
=

2
cos
dx
1
2
2
=
+

; 
11) 
C
ctgx
x
x
arctg
a
x
a
; 
12) 
+C
a
dx
+
−
=

2
sin
a
x
1
x
dx
+
=
2
2
; 
dx
+
+
−
=
13) 
C
a
x
2
2
; 
14) 
C
a
−

ln
2
−

arcsin
a
a
x
x
a
2
2
dx
+

+
=
15) 
C
a
x
x
2
2
ln
; 


a
x
dx
=arcsinx + C= – arccosx +C; 
 
16) 
−
2
1
x
dx
=arctgx + C = –arcctgx +C.  
17) 
+1
2
x
Example. To solve   
1.
(
)

+
+
dx
x
x
x
cos
5
2 2
3
. 
Solution: From the properties of integrals we have  
(
)

+
+
dx
x
x
x
cos
5
2 2
3
= 
=


+
+
xdx
dx
x
dx
x
cos
5
2
2
3
3
4
2
;
sin
5
C
x
x
x
+
+
+
=
 
4
3
Another examples. Find the following integrals  
2. (
)
C
x
x
x
C
x
x
x
dx
x
x
+
+
+
=
+
+

+

=
+
+

7
2
7
4
4
3
3
7
4
3
2
3
2
3
2
; 
6 




3
3
2
3
2
3. 

=
3
8
6
7
8
6
7
 

+
+
−
dx
x
x
x
dx
x









+
+
=



x
x


2
2
3
5
C
x
x
x
C
x
x
x
3
5
24
3
ln
7
=
+
+
−
=
+

+
−

+
=
−
−
5
3
5
8
2
6
ln
7
 
x
+
+
−
=
3
5
C
x
24
3
ln
7
; 
2
5
x




−
+
+
dx
x
x
dx
x
x
x
cos
3
2
9
cos
3
2
9
2
3
4. 

=


+
+
=


3
 






x




−
x
2
1
2
C
x
=
+
+
+
2
18
; 
C
x
x
x
+
+
+
−
=
sin
3
2
ln
sin
3
2
ln
x
−
2
1
9

+
−
C
x
x
dx
tgx
x
cos
ln
5
cos
; 
3
2
3
2
5
sin



5. 
+
−
=




4
6
+
C
arctgx
x
dx
2
2
; 
+
1
5
−
x
x
5
6
arcsin
4
1
6. 
(
)

+
+
=






dx
8
2
; 
7. 
+
+
−
=
C
x
x
7
7
ln
7
4
−
x
49
2
2


5
4
1
+
+
=
+
=
; 
+
C
arctgx
x
dx
x
dx
dx
x
4
4
1
2
2
2
+
+
+
8. 



+
+
=






x
x
x
1
1
1
2
−
1
x
 We can rewrite this integral by mathematical formu
x
9. (
) .
4
las and so we get that, 
7 

x
x
x
4
/
1
4
/
1
4
/
3
2
1
2
1
 
(
)
(
)



=
+
−
=
+
−
=
−
−
dx
x
x
x
dx
4
/
1
4
x
x
4
4
/
3
4
/
5
4
/
7
4
4
2
.
C
x
x
x
+
+
−
=
 
7
5
3
2
2
x
x
3
3
10. 
=
 
+
−
−

dx
4
−
x
9


2
2
x
x
3
3
+
−
=
−
= 
dx
2
2
2
2






+
−
+
−
x
x
x
x
3
3
3
3
(
)(
)
(
)(
)




1
1
−
=
dx



=


2
2
C
x
x
x
x
+
−
−
−
+
+
3
ln
3
ln
2
2
; 
+
−
3
x
x
3


x
10
x
x
x
x






10
5
2
5
2
; 
11. 


+
x
e
dx
e
dx
e
dx
=






=







=

C
e
10
ln
e
2
cos
1
2
; 
12. 
C
x
dx
x
dx
x
x
dx
x
x
+
−
=
=
=
−



cos
2
sin
2
sin
sin
2
sin
dx
=
dx
+
=
+
=
13. 
C
x
C
x


2
3
arcsin
3
1
−
x
9
4
2
2
. 
−
x
3
2
arcsin
3
1
9
4
3
1
Do the following exercises and test Yourself 
1. dx
4
;  
  2. 
dt
t2
3
; 
3. 
−dx
x 3
2
; 
  4. ds ; 
8 

5. 
dx
x 4
21
; 
  6. 
dx
x
5
12
; 
du
dx
; 
7. 
3
2x
; 
  8. 
2
4u
dx ; 
  10. (
)

+
dx
x
6
4 3
; 
9. 5
3
x


−
+
dx
x
x
1
4
3
4
; 



11. (
)

−
+
dx
x
x
2
6
5
2
4
; 
  12. 





+1
2 2
+
dt
t
)
4
(
; 
  14. 
du
; 
u

13. 
u
t
+
3
15. 
dz
z

; 
  16. 
(
)

+
dx
x
x
4
3
; 
2
z
17. (
)

+
du
u
u
2
4 2
; 
  18. 
(
)

−
−
dx
x
x
5
6
)
1
(
; 

4

−
dx
ctgx
4
2
; 
19. 
dx
x
x2
7
;  
  20. 






x
cos



5
3
11
−

+
+
−
dx
e
x
x
2
3
dx
2
;   22. 




2
2
; 
21. 






x
sin
2
+
−
x
x
9
5


dx
4
dx
5
; 
23. 
2
x
+ 49
x
−
2
16
2
; 
  24. (
)

x
x
4
3
2
+
−
dx
dx
; 
25. 
; 
  26. 
x
x
−
2
3
7
x
9 

+
−
2
4
3
1
x
x
x
4
7
13
4
5
4
+
−
dx
27. 
dx
;
. 
x
x

28. 
x
x
  
 
Answers:  
1) 
C
x +
4
 ;     
2) 
C
t +
3
;     
3) 
C
x
+
−−2
 ;     
4) 
C
s +
;   
5) 
C
x
+
4
7
12
;     
6) 
C
x
x
+
5
10
;     
1
7) 
C
1
 ;    
;      
8) 
C
u +
−4
x
+
−
2
4
55
2
;     
10) 
C
x
x
+
+6
4
;      
9) 
C
x
+
2
3
;     
11) 
C
x
x
x
+
−
+
2
2 3
5
;  
12) 
C
x
x
x
+
−
+
2
3
7
2
7
13) 
C
t
t
+
+
|
|
ln
4
;      
14) 
C
u
u
+
+
|
|
ln
2
;   
15) 
C
z
z
+
−3
|
|
ln
;       
16) 
C
x
x
+
+
2
3
2
;     
11
2
2
3
;     
17) 
C
u
u
+
+
2
4
;     
18) 
C
x
x
x
+
+
−
5
2
19) 
C
x
x
+
3
2
;   
20) 
C
x
tgx
+
−
sin
ln
4
4
;   
3
21) 
C
e
ctgx
x
x
;    
2
5
+
+
+
2
3
ln
10