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Файл:Mathematics for Foreign Students integrals. Study Guide
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91
4) ; 5) .
6.
1) ; 2) x+2arcsin(x–2)+C;
3) ;
4) ; 5) arctg(x–2)+C.
7. The area which is bounded by the lines of functions
and , is equal to:
1) 4; 2) 2; 3) 2,5; 4) 3; 5) 6.
8. The length of the arc: х=cos2t, y=sin2t, in the interval
t0
, is
equal to:
1) 1/2 π; 2) 2 π; 3) 4 π; 4) 4 π2; 5) 1.
9. The volume of the solid, which is generated by revolving the region
around the x-axis, where the region is bounded by y=4–x, y=0 and х=0, is
equal to:
1)
3
2
; 2)
3
64
; 3) 9 π; 4)
3
8
; 5) 2 π.
10. Solving the following improper integral
+
−
+
=
2
1 х
dx
I
leads to the next result:
1) Integral diverges; 2) I=2 π; 3) I= π;
4) Integral converges; 5) no solutions.
ТЕSТ № 5
1. Choose the correct answer for the integral
dx
x
xsin
+−
2
3
4
32
1)
3
4
2
4
2
3
3
−
++
lnln
; 2)
3
4
2
4
2
3
3
−
+−
lnln
;
cos5 cos
10 2
xx
C− − +
cos5 cos
10 2
xx
C++
2
( 2)
45
х dx
хx
+
−+
2
45х x C− + +
2
4 5 4ln| 2 |х x x C− + + − +
22
4 5 4ln | 2 4 5 |х x x х x C− + + − + − + +
2
24y x x= − +
2
54y x x= − + +
92
3)
3
4
2
4
2
3
6
−
+−
lnln
; 4)
3
4
2
4
2
3
6
−
++
lnln
;
5)
3
4
2
4
2
3
3
+
+−
lnln
.
2. Choose the correct answer for the integral
1) ; 2) ; 3) 8;
4) ; 5) .
3. Choose the correct answer for the integral
xdxsinx
2
1) ; 2) ;
3) ; 4) ;
5) .
4. Choose the correct answer for the integral
−
+
dx
xx
x
23
2
1)
C|x|ln
x
|x|ln +−++− 13
2
3
; 2)
C|x|ln|x|ln +−+− 133
;
3)
C|x|ln
x
|x|ln +−+− 13
2
3
; 4)
C|x|ln
x
|x|ln +−+−− 13
2
3
;
5)
C|x|ln|x|ln +−+ 133
.
5. Choose the correct answer for the integral
dx
xsin
xsin
+
2
0
2
21
2
1) 4; 2)
23−
; 3)
12
1
; 4)
4
1
; 5)
21−
.
6. Choose the correct answer for the integral
+
9
1
14 xx
dx
3
4
1
2
1
xdx
x+
ln 3 10 ln 1 2+ − +
arctg9
4
−
arctg3
4
−
ln 9 82 ln 1 2+ − +
( )
2
2 cos 2 sinх x x x C− + +
3
cos
3
x
xC+
( )
2
2 cos 2 sinх x x x C− + +
3
cos
3
x
xC−+
( )
2
2 cos 2 sinх x x x C− − +
93
1)
26−
; 2)
23−
; 3)
22−
;
4)
222 −
; 5)
21−
.
7. The area which is bounded by the hyperbola xy=9, the axis Oх, and
the lines x=3 and x=6, is equal to
1) 9ln2; 2) 9ln3; 3) ln2; 4) ln3; 5) 9.
8. The length of the cardioid arc =2(1+cos) from x1=0 to x2= π, is
equal to:
1) 8; 2) 16; 3) 15; 4) 0; 5) π.
9. The volume of the solid, which is generated by revolving the region
around the y-axis, where the region is bounded by y=2x
2
,
y=1 and х=0, is
equal to:
1)
12
1
; 2)
4
1
; 3) 4; 4)
3
8
; 5)
16
.
10. Solving the following improper integral
dx
x
x
I
−
=
6
1
2
1
leads to
the next result:
1) Integral diverges; 2) I=2; 3) I=2,5;
4) Integral converges; 5) no solutions.
ANSWERS
1 2 3 4 5 6 7 8 9 10
1 5 1 3 5 1 3 4 4 2 1
2 1 5 1 3 5 2 3 4 1 5 3 4 3 2 3 1 4 3 3 4 1 4 3 1 5 1 2 4 1 3 2 3 5 2 5 3 1 4 3 1 1 2 1
94
ТЕSТ № 6
1.
+ xdxx)( cossin1
.
2.
xdxхsin
.
3.
+− 65
2
xx
dx
.
4.
xdx
2
cos
.
5.
−
3
13x
dx
.
6.
( )
−
−−
1
3
2
132 dxxx
.
7. Find the area which is bounded by lines y=x2 +2, y=0, x= –2, x=1.
8. Find the volume of the solid which is generated by revolving the region
which is bounded by y=(x–1)
2
+2, y=0, x= 0 and x=2, around the axis.
9. Find the length of arc of the following curve y=x2, from the point
A(1;1) to B(2;4).
ТЕSТ № 7
1.
+ dxx3
.
2.
dxxe
x
3.
−−−
−
)12)(1(
)6743(
2
xxx
dxx
4.
xdxx
23
cossin
.
5.
−+− 165
2
xx
dx
.
6.
0
cos xdxx
.
7. Find the area which is bounded by the lines of functions y=
5
1
x3,
y=x2 and x=3.
95
8. Find the volume of the solid which is generated by revolving the
region which is bounded by y=2x–x
2
, y=0, around the axis.
9. Find the length of arc of the curve x=
4
1
y2 –
2
1
lny, in the interval
у=1 and у=2.
ТЕSТ № 8
1.
−dxe
e
x
x
7
.
2.
+ xdxх cos)43(
.
3.
+−
+−
)4()1(
)522(
22
23
xx
dxxx
4.
dx
x
x
3
2
sin
cos
.
5.
−+ 43
2
xx
dx
6.
8
0
3
2
x
dx
.
7. Find the area which is bounded by line y=x2 – 4 and the axis Оx.
8. Find the volume of the solid which is generated by revolving the region
which is bounded by y2=4x, and the interval 0 ≤ х ≤5, around the axis Оx.
9. Find the length of arc of the curve у=x
2/3
, in the interval 0 ≤ х ≤ 5.
ТЕSТ № 9
1.
+ dxx )13sin(
.
2.
xdxxcos
3.
−−
−
)1)(1(
)3(
2
3
xx
dxx
.
4.
xdxx 5cos4sin
.
96
5.
+− 136
2
xx
dx
.
6.
2
0
2
cos
xdx
.
7. Find the area which is bounded by the function y=x3 – 4х and the
axis Оx.
8. Find the volume of the solid which is generated by revolving the
region which is bounded by у= x2 and the axis Оy, around the axis Оy.
9. Find the length of arc of the curve r=a(1–cosx).
ТЕSТ № 10
1.
− dxxx
5
)1(
.
2.
arctgxdx
.
3.
++−
+
)136)(1(
)23(
2
2
xxx
dxx
.
4.
xdxx cos
2
.
5.
−+
2
5 xx
xdx
.
6.
4
0
2
xdxxtg
.
7. Find the area which is bounded by the lines у= – x3, у=х–1, у=0, у=1.
8. Find the volume of the solid which is generated by revolving the
region which is bounded by 2х–y–2=0, y=0, x=3, around the axis Оx.
9. Find the length of arc of the curve x=acos3 t, y=asin3 t.
R E F E R E N C E S
1. Математика: учебное пособие / Ю. М. Данилов,
Л. Н. Журбенко, Г. А. Никонова [и др.]. – М.: Инфра-М, 2016. – 495 с.
2. Математика в примерах и задачах: учеб. пособие /
Л. Н. Журбенко, Г. А. Никонова, Н. В. Никонова [и др.]. – М.: ИнфраМ, 2016. – 372 с.
3. Данко, П. Е. Высшая математика в упражнениях и задачах.
Ч. 1 / П. Е. Данко, А. Г. Попов, Т. Я. Кожевникова. – М.: Высшая школа,
1999. – 282 с.
4. Шнейдер, В. Е. Краткий курс высшей математики. Т. 1 /
В. Е. Шнейдер, А. И. Слуцкий, А. С. Шумов. – М.: Высш.школа, 1978.–
384 с.
97
F O R N O T E S
98
EDUCATIONAL EDITION
Dilbar Bikmukhametova, Natalya Gazizova, Svetlana Enikeeva,
Alsu Mindubaeva, Natalia Nikonova, Serazh Alkhaleefah
MATHEMATICS FOR FOREIGN
STUDENTS: INTEGRALS
Responsible for the publication A. Yegorov
Дильбар Наилевна Бикмухаметова, Наталья Николаевна Газизова,
Светлана Рашидовна Еникеева, Алсу Рафаэлевна Миндубаева,
Наталия Владимировна Никонова, Сераж Алхалифах
МАТЕМАТИКА ДЛЯ ИНОСТРАННЫХ
СТУДЕНТОВ: ИНТЕГРАЛЫ
99
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