T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
64-son_3-to’plam_Iyun-2025
118
ISSN:3030-3613
TARKIBIDA TRIGONOMETRIK FUNKSIYALAR QATNASHGAN
IFODALARNI INTEGRALLASH.
Termiz davlat universiteti
“Axborot texnologiyalari” fakulteti
“Amaliy matematika” ta’lim yo‘nalishi
bakalavr II bosqich talabasi
Xolturayeva Kamola Bahrom qizi
Annotatsiya:
Biz bu maqolada tarkibida trigonometrik funksiyalar qatnashgan
ifodalarni sodda ko`rinishda integralashni o`rgamiz. Ya`ni bu maqolada trigonometrik
funksiyarlarni belgilash kiritish yo`li bilan qulay usulda ishlashni o`rganamiz.Bundan
tashqari maple dasturida trigonometrik ifodalarning grafigini chiqarishni ham
o`rganamiz.
Kalit
so`zlar:
Ratsional
funksiya,integral,trigonometrik
funksiya,toq
funksiya,juft funksiya.
Аннотация:
В данной статье рассматривается интегрирование выражений,
содержащих тригонометрические функции, в упрощённом виде. Мы изучаем
удобные методы работы с тригонометрическими функциями с помощью
введения обозначений. Кроме того, рассматривается построение графиков
тригонометрических выражений с использованием программы Maple.
Ключевые слова:
рациональная функция, интеграл, тригонометрическая
функция, нечётная функция, чётная функция.
Abstract:
In this article, we learn to integrate expressions involving
trigonometric functions in a simplified form. Specifically, we explore convenient
methods of working with trigonometric functions by introducing appropriate notations.
Additionally, we study how to plot graphs of trigonometric expressions using the
Maple software.
Keywords:
rational function, integral, trigonometric function, odd function,
even function.
Hamma trigonometrik funksiyalarni integrallash mumkin.Bu ifodani
𝑅(𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥)
orqali
belgilaymiz.Endi
𝑅(𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥)
ko`rinishidagi
ifodani
integrallaymiz.
∫ 𝑅(𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥)𝑑𝑥
Bunday integralni
𝑡𝑔
𝑥
2
= 𝑡
belgilash yordamida
𝑡
o`zgaruvchili ratsional
funksiyaning integraliga almashtirish mumkin.Integralni bunday almashtirish
ratsionallashtirish deyiladi. Ko`rinib turibdiki,
𝑡𝑔
𝑥
2
= 𝑡
desak,
T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
64-son_3-to’plam_Iyun-2025
119
ISSN:3030-3613
𝑠𝑖𝑛𝑥 =
2𝑡𝑔
𝑥
2
1+𝑡𝑔
2𝑥
2
=
2𝑡
1+𝑡
2
;
𝑐𝑜𝑠𝑥 =
1−𝑡𝑔
2𝑥
2
1+𝑡𝑔
2𝑥
2
=
1−𝑡
2
1+𝑡
2
;
𝑥
2
= 𝑎𝑟𝑐𝑡𝑔𝑡 ≫ 𝑥 =
2𝑎𝑟𝑐𝑡𝑔𝑡 ≫ 𝑑𝑥 =
2𝑑𝑡
1+𝑡
2
Shuning uchun
∫ 𝑅(𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥)𝑑𝑥
=
∫ 𝑅 (
2𝑡
1+𝑡
2
;
1−𝑡
2
1+𝑡
2
)
2𝑑𝑡
1+𝑡
2
= ∫ 𝑅
1
(𝑡)𝑑𝑡
bunda
𝑅
1
(𝑡) − 𝑡
o`zgaruvchili ratsional funksiya.Bunday almashtirish
𝑅(𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥)
ko`rinishidagi har qanday funksiyani integrallashga imkon beradi,
shuning uchun bunday almashtirish ko`pincha ancha murakkab ratsional funksiyaga
olib keladi.
1
0
.
Agar
𝑅(𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥)
ifoda
′′𝑠𝑖𝑛𝑥
′′
ga nisbatan toq funksiya,ya`ni
𝑅(−𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥) = −𝑅(𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥)
bo`lsa,
u
holda
′′𝑐𝑎𝑠𝑥 = 𝑡
′′
𝑥 ∈ (0; 𝜋)
almashtirish bajarilsa
(1; 1)
integral ostidagi ifoda
𝑡
ning ratsional funksiyasiga
keltiriladi.
1-misol.
∫ 𝑠𝑖𝑛
3
𝑥 ∙ 𝑐𝑜𝑠
3
𝑥𝑑𝑥 = |
𝑐𝑜𝑠𝑥 = 𝑡
𝑥 = 𝑎𝑟𝑐𝑐𝑜𝑠𝑡
𝑠𝑖𝑛
3
𝑥 = √(1 − 𝑐𝑜𝑠
2
𝑥)
3
𝑑𝑥 = −
𝑑𝑡
√1−𝑡
2
|
=
∫
√(1−𝑡
2
)
3
𝑡
2
(−1)𝑑𝑡
√1−𝑡
2
= ∫(1 − 𝑡
2
)𝑡
2
𝑑𝑡 =
𝑡
5
5
−
𝑡
3
3
=
1
5
𝑐𝑜𝑠
5
𝑥 −
1
3
𝑐𝑜𝑠
3
𝑥 + 𝐶
>
f:=(((sin(x))^3)*((cos(x))^3));
>
Int (f,x);
>
Int(f,x)=int (f,x);
>
F:plot((((sin(x))^3)*((cos(x))^3)), x=-Pi..Pi, y=-Pi..Pi);
:=
f
( )
sin
x
3
( )
cos
x
3
d
( )
sin
x
3
( )
cos
x
3
x
d
( )
sin
x
3
( )
cos
x
3
x
1
6
( )
sin
x
2
( )
cos
x
4
1
12
( )
cos
x
4
T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
64-son_3-to’plam_Iyun-2025
120
ISSN:3030-3613
Demak, bu integralning chizmasi yuqoridagi ko`rinishda ekan.
2
0
. Agar
𝑅(−𝑠𝑖𝑛𝑥, −𝑐𝑜𝑠𝑥) = 𝑅(𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥)
bo`lsa, u holda
𝑡 = 𝑡𝑔𝑥
𝑥 ∈ (
𝜋
2
; −
𝜋
2
)
almashtirishdan biri bajariladi.
2-misol.
∫
3𝑠𝑖𝑛𝑥−2𝑐𝑜𝑠𝑥
1+𝑐𝑜𝑠𝑥
𝑑𝑥 = |
𝑡𝑔
𝑥
2
=𝑡 𝑥=2𝑎𝑟𝑐𝑡𝑔𝑡 𝑐𝑜𝑠𝑥=
1−𝑡2
1+𝑡2
𝑑𝑥=
2
1+𝑡2
𝑠𝑖𝑛𝑥=
2𝑡
1+𝑡2
|
=
∫
2(3
2
1+𝑡2
−2
1−𝑡2
1+𝑡2
)
(1+
1−𝑡2
1+𝑡2
)(1+𝑡
2
)
𝑑𝑡 =
2 ∫
𝑡
2
+1−2+8𝑡
1+𝑡
2
𝑑𝑡 = 2 ∫ 𝑑𝑡 − 4 ∫
𝑑𝑡
1+𝑡
2
+ 6 ∫
𝑑(1+𝑡
2
)
1+𝑡
2
= 2𝑡 − 4𝑎𝑟𝑐𝑡𝑔𝑡 +
+3𝑙𝑛|1 + 𝑡
2
| + 𝐶
=2tg(
𝑥
2
) + 3 ln (𝑡𝑔 (
𝑥
2
)
2
+ 1) − 2𝑥 + 𝐶;
Endi bu misolni yechimini va grafigini maple dasturida ko`ramiz:
>
f:=((3*sin(x)-2*cos(x))/(1+cos(x)));
>
Int(f,x);
>
int(f,x);
>
Int(f,x)=int(f,x);
>
plot(((3*sin(x)-2*cos(x))/(1+cos(x))), x=-4..4, y=-4..4);
:=
f
3
( )
sin
x
2
( )
cos
x
1
( )
cos
x
d
3
( )
sin
x
2
( )
cos
x
1
( )
cos
x
x
2
tan
x
2
3
ln
tan
x
2
2
1
2
x
d
3
( )
sin
x
2
( )
cos
x
1
( )
cos
x
x
2
tan
x
2
3
ln
tan
x
2
2
1
2
x
T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
64-son_3-to’plam_Iyun-2025
121
ISSN:3030-3613
Demak, bu integralning chizmasi yuqoridagi ko`rinishda ekan.
3
0
.
∫ 𝑠𝑖𝑛𝛼 ∙ 𝑥𝑐𝑜𝑠𝛽𝑥 𝑑𝑥 ,
∫ 𝑠𝑖𝑛𝛼 ∙ 𝑥𝑠𝑖𝑛𝛽𝑥 𝑑𝑥
,
∫ 𝑐𝑜𝑠𝛼𝑥 ∙ 𝑐𝑜𝑠𝛽𝑥 𝑑𝑥
, ko`rinishidagi
integrallarni hisoblash.
Foydalanuvchi formulalar:
1) 𝑠𝑖𝑛𝛼𝑥 ∙ 𝑐𝑜𝑠𝛽𝑥 =
1
2
[sin(𝛼 + 𝛽) 𝑥 + sin(𝛼 − 𝛽)𝑥]
;
2) 𝑠𝑖𝑛𝛼𝑥 ∙ 𝑠𝑖𝑛𝛽𝑥 =
1
2
[cos(𝛼 − 𝛽)𝑥 − cos(𝛼 + 𝛽) 𝑥];
3) 𝑐𝑜𝑠𝛼𝑥 ∙ 𝑐𝑜𝑠𝛽𝑥 =
1
2
[cos(𝛼 + 𝛽)𝑥 + cos(𝛼 − 𝛽) 𝑥];
3-misol.
∫ 𝑐𝑜𝑠4𝑥 ∙ 𝑐𝑜𝑠𝑥𝑑𝑥 =
1
2
(𝑐𝑜𝑠5𝑥 − 𝑐𝑜𝑠3𝑥)𝑑𝑥 =
1
2
∫ 𝑐𝑜𝑠5𝑥 𝑑𝑥 +
1
2
∫ 𝑐𝑜𝑠3𝑥 𝑑𝑥 =
1
10
𝑠𝑖𝑛5𝑥 +
1
6
𝑠𝑖𝑛3𝑥 + 𝐶.
>
f:=((cos(4*x)*cos(x)));
>
Int(f,x);
>
int(f,x);
>
Int(f,x)=int(f,x);
>
plot(((cos(4*x)*cos(x))), x=-4..4, y=-4..4);
:=
f
(
)
cos 4
x
( )
cos
x
d
(
)
cos 4
x
( )
cos
x
x
1
6
(
)
sin 3
x
1
10
(
)
sin 5
x
d
(
)
cos 4
x
( )
cos
x
x
1
6
(
)
sin 3
x
1
10
(
)
sin 5
x
T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
64-son_3-to’plam_Iyun-2025
122
ISSN:3030-3613
Demak, bu integralning chizmasi yuqoridagi ko`rinishda ekan.
4
0
.
∫ 𝑠𝑖𝑛
𝑚
𝑥 ∙ 𝑐𝑜𝑠
𝑛
𝑥𝑑𝑥 (𝑛, 𝑚 ∈ 𝑧)
ko`rinishidagi integrallarni hisoblash.
𝐼.
𝑛, 𝑚
lar manfiy bo`lmagan
(𝑛, 𝑚 ∈ 𝑧) 𝑛 > 0, 𝑚 > 0
juft son bo`lgan holat.Bu
holatda darajani pasaytirish, ya`ni
𝑠𝑖𝑛
2
𝑥 =
1
2
(1 − 𝑐𝑜𝑠𝑥) 𝑐𝑜𝑠
2
𝑥 =
1
2
(1 + 𝑐𝑜𝑠2𝑥)
Formula qo`llaniladi.
4-misol.
∫ 𝑠𝑖𝑛
4
𝑥 ∙ 𝑐𝑜𝑠
2
𝑥𝑑𝑥 =
1
4
∫ 𝑠𝑖𝑛
2
𝑥 ∙ 𝑠𝑖𝑛
2
2𝑥𝑑𝑥 =
1
4
∫
1−𝑐𝑜𝑠2𝑥
2
∙
1−𝑐𝑜𝑠4𝑥
2
𝑑𝑥 =
1
16
∫(1 − 𝑐𝑜𝑠4𝑥 − 𝑐𝑜𝑠2𝑥 + 𝑐𝑜𝑠2𝑥 ∙ 𝑐𝑜𝑠4𝑥)𝑑𝑥 =
1
16
𝑥 −
1
64
𝑠𝑖𝑛4𝑥 −
1
32
𝑠𝑖𝑛2𝑥 +
1
16
∫(𝑐𝑜𝑠2𝑥 + 𝑐𝑜𝑠6𝑥)𝑑𝑥 =
1
16
𝑥 −
1
64
𝑠𝑖𝑛2𝑥 −
1
64
𝑠𝑖𝑛4𝑥 +
1
182
𝑠𝑖𝑛6𝑥 + 𝐶.
𝐼𝐼. 𝑛, 𝑚
larning ikkalasi ham butun manfiy
𝑛, 𝑚 ∈ 𝑧
va juft yoki toq bo`lganda
𝑡𝑔𝑥 =
𝑡
yoki
𝑐𝑡𝑔𝑥 = 𝑡
olinib
1 + 𝑡𝑔
2
𝑥 =
1
𝑐𝑜𝑠
2
𝑥
1 + 𝑐𝑡𝑔
2
𝑥 =
1
𝑠𝑖𝑛
2
𝑥
formuladan
foydalaniladi.
5-misol.
∫
1
𝑠𝑖𝑛
3
𝑥∙𝑐𝑜𝑠
6
𝑥
𝑑𝑥 = ∫
1
𝑠𝑖𝑛
8
𝑥∙
𝑐𝑜𝑠5𝑥
𝑠𝑖𝑛5𝑥
= ∫
𝑑𝑥
𝑠𝑖𝑛
2
𝑥∙𝑠𝑖𝑛
6
𝑥∙
𝑐𝑜𝑠5𝑥
𝑠𝑖𝑛5𝑥
= − ∫
1
𝑐𝑡𝑔
5
𝑥
∙
(1 + +𝑐𝑡𝑔
2
𝑥)
3
𝑑(𝑐𝑡𝑔𝑥) = |
𝑐𝑡𝑔𝑥=𝑡
𝑑𝑐𝑡𝑔𝑥=𝑑𝑡
| =
1
4
𝑡
−4
+
3
2
𝑡
−2
− 3 ln|𝑡| −
𝑡
2
2
+ 𝐶 =
1
4
∙
1
𝑐𝑡𝑔
4
𝑥
+
3
2
∙
1
𝑐𝑡𝑔2𝑥
− 3𝑙𝑛|𝑐𝑡𝑔𝑥| +
𝑐𝑡𝑔
2
𝑥
2
+ 𝐶.
>
f:=(1/((sin(x))^3)*((cos(x))^6));
>
Int (f,x);
:=
f
( )
cos
x
6
( )
sin
x
3
T A D Q I Q O T L A R
jahon ilmiy – metodik jurnali
https://scientific-jl.com
64-son_3-to’plam_Iyun-2025
123
ISSN:3030-3613
>
Int(f,x)=int (f,x);
>
F:plot((1/((sin(x))^3)*((cos(x))^6)), x=-Pi..Pi, y=-Pi..Pi);
Xulosa:
Trigonometrik ifodalarni soda ko`rinishga keltirmasdan uni integrallash
ancha murakkab ishligi sababli uni yechishni ossonlashtirdik.Ya`ni belgilash kiritish
orqali integrallarga yechim topdik.Sodda ko`rinishga keltirilganligi sababli
trigonometrik ifodalarni integralllash ancha oson bo`ladi. Maple 9.5 dasturidan
foydalanib, berilgan trigonometrik ifodalarnni integrallashni yechim qildik va
yechimlarni grafiklari bilan ham tanishdik.
Foydalanilgan adabiyotlar ro’yxati:
1.
Azlarov T.A, Mansurov X.T. Matematik analiz. 1-qism.-T.:<<O`qituvchi>> 1994.
2.
Azlarov T.A., Mirzahmedov M.A. Otaqo`ziyev D.O., Sobirov M.A.,To`laganov
S.T.-Matematikadan qo`llanma, II qism. T.:<<O`qituvchi>> 1990.
3.
Gaziyev A., Israilov L.,Yaxshiboyev M. Funksiyalar va grafiklar.-T:”VORIS-
NASHRIYOT” , 2006.
4.
Демидович Б.П. сборник задач и упражиский по математическому анализу, -
М.: Паука, 1981.
5.
Sadullayev A., Mansurov X., Xudoyberganov G., Vorisov A., G`ulomov
R.Matematik analiz kursidan misol va masalalar to`plami.1-qism. –T., 1993.
d
( )
cos
x
6
( )
sin
x
3
x
d
( )
cos
x
6
( )
sin
x
3
x
1
2
( )
cos
x
7
( )
sin
x
2
1
2
( )
cos
x
5
5
6
( )
cos
x
3
5
2
( )
cos
x
5
2
(
)
ln
( )
csc
x
( )
cot
x
