Ths Đ ng Thanh C u-01696900100 ặ ầ BÀI T P NGUYÊN HÀM TÍCH PHÂNẬ
I. Tìm nguyên hàm b ng đ nh nghĩa và các tính ch tằ ị ấ
1/ Tìm nguyên hàm c a các hàm s .ủ ố
1. f(x) = x2 – 3x +
x
1
2. f(x) =
2
432
x
x+
3. f(x) =
4
3xxx ++
4. f(x) =
2
22 )1(
x
x−
5. f(x) =
2
1
x
x−
6. f(x) =
3
21
xx −
7. f(x) =
x
x2
)1( −
8. f(x) =
3
1
x
x−
9. f(x) =
2
sin2 2x
10.f(x) = tan2x 11. f(x) = cos2x 12. f(x) = (tanx – cotx)2 13. f(x) =
xx 22 cos.sin
1
14. f(x) =
xx
x
22 cos.sin
2cos
15.f(x) = sin3x f(x) = 2sin3xcos2x 17. f(x) = ex(ex – 1)
18. f(x) = ex(2 +
)
cos2x
ex−
19. f(x) = 2ax + 3x 20. f(x) = e3x+1
2/ Tìm hàm s f(x) bi t r ng ố ế ằ
1. f’(x) = 2x + 1 và f(1) = 5 2. f’(x) = 2 – x2 và f(2) = 7/ 3. f’(x) = 4
và f(4) = 0
4. f’(x) = x -
2
1
2+
x
và f(1) = 2 5. f’(x) = 4x3 – 3x2 + 2 và f(-1) = 3
6.f’(x) = ax +
2)1(,4)1(,0)1(',
2=−== fff
x
b
ĐS. f(x) =
2
51
2
2++ x
x
II. M T S PH NG PHÁP TÌM NGUYÊN HÀMỘ Ố ƯƠ
1.Ph ng pháp đ i bi n s .ươ ổ ế ố
Tính I =
∫dxxuxuf )(')].([
b ng cách đ t t = u(x)ằ ặ
Đ t t = u(x)ặ
dxxudt )('=⇒
I =
∫ ∫
=dttfdxxuxuf )()(')].([
BÀI T PẬ
Tìm nguyên hàm c a các hàm s sau:ủ ố
1.
15
(5 1)x dx−
2.
12
( 2)x x dx−
3.
∫−5
)23( x
dx
4.
dxx
∫−25
5.
∫−12x
dx
6.
∫+xdxx 72 )12(
7.
∫+dxxx 243 )5(
8.
xdxx .1
2
∫+
9.
∫+dx
x
x
5
2
10.
( )
5
2
xdx
x−
11.
dx
x
x
∫3
ln
12.
∫+2
)1( xx
dx
13.
∫+dx
x
x
3
2
25
3
14.
∫+dxex x1
2
.
15.
∫xdxx cossin 4
16.
cot xdx
17.
∫dx
x
x
5
cos
sin
18.
∫x
tgxdx
2
cos
19.
∫xdxx 23 sincos
20.
21.
∫tgxdx
22.
∫dx
x
ex
23.
∫−3
x
x
e
dxe
24.
∫dx
x
etgx
2
cos
25.
∫
−dxx .1
2
26.
∫−2
4x
dx
27.
∫−dxxx .1 22
28.
∫+2
1x
dx
29.
∫−2
2
1x
dxx
30.
∫++ 1
2xx
dx
31.
∫x
dx
sin
32.
dxxx .1
∫−
33.
∫+1
x
e
dx
34.
dxxx .1
23
∫+
35.
3
sin
dx
x
36.
3
os
dx
c x
37.
3
tan
dx
x
38.
3
osc xdx
39.
3
sin xdx
40.
(sinx+ cos )
sinx cos
x dx
x−
41.
2
2 2
dx
x x+ +
42.
sin 4 sinx xdx
43.
3
sin
cos
xdx
x
44.
1
( ln )
x
x
xe dx
x e x
+
+
45.
3sin cosx xdx
46.
cos3 sinx xdx
47.
2
3 1
dx
x+
48.
2 3
(1 )
dx
x+
49.
3 2
1x x dx−
50.
21
dx
x x +
51.
2
(2 1)
xdx
x+
52.
3
1 3
x
xdx
−
53.
2
2 1
x dx
x+
54.
41
dx
x x +
55.
3
1 3
x
xdx
−
56.
21
dx
x x −
57.
33 2
1x x dx−
1
Ths Đ ng Thanh C u-01696900100 ặ ầ BÀI T P NGUYÊN HÀM TÍCH PHÂNẬ
2. Ph ng pháp l y nguyên hàm t ng ph n.ươ ấ ừ ầ
N u u(x) , v(x) là hai hàm s có đ o hàm liên t c trên Iế ố ạ ụ
∫ ∫
−= dxxuxvxvxudxxvxu )(').()().()(').(
Hay
∫ ∫
−= vduuvudv
( v i du = u’(x)dx, dv = v’(x)dx)ớ
Tìm nguyên hàm c a các hàm s sau:ủ ố
1.
∫xdxx sin.
2.
∫xdxx cos
3.
∫xdxx ln
4.
∫xdxln
5.
∫xdxx 2sin
6.
∫dxex x
.
7.
∫xdxx 2cos
8.
∫++ xdxxx cos)32( 2
9.
∫+xdxx sin)5( 2
10.
dxx
∫2
ln
11.
∫dxe x
12.
∫dxxsin
13.
∫dx
x
x
2
cos
14.
∫+dx
x
x
2
)1ln(
15.
∫x
xdxln
6.
∫+dxx )1ln( 2
17.
∫xdxexcos.
18.
∫dxex x2
3
19.
∫+dxxx )1ln( 2
20.
∫xdx
x
2
21.
∫xdxx lg
22.
∫+dxxx )1ln(2
23.
∫xdxxtg 2
24.
∫xdxx 2cos
2
TÍCH PHÂN
I. TÍNH TÍCH PHÂN B NG CÁCH S D NG TÍNH CH T VÀ NGUYÊN HÀM C B N:Ằ Ử Ụ Ấ Ơ Ả
1.
1
3
0
( 1)x x dx+ +
2.
2
2
1
1 1
( )
e
x x dx
x x
+ + +
2.
3
1
2x dx−
3.
2
1
1x dx+
4.
1
0
( )
x
e x dx+
5.
2
3
(2sin 3 )x cosx x dx
π
π
+ +
6.
1
3
0
( )x x x dx+
7.
2
1
( 1)( 1)x x x dx+ − +
8.
2
3
1
(3sin 2 )x cosx dx
x
π
π
+ +
9.
1
2
0
( 1)
x
e x dx+ +
10.
2
23
1
( )x x x x dx+ +
11.
3
3
1
x 1 dx( ).
−
+
12.
2
1
( 1)( 1)x x x dx− + +
13.
2+
2
2
-1
x.dx
x
14.
2
e
1
7x 2 x 5 dx
x
− −
15.
x 2+ + −
5
2
dx
x 2
16.
2
2
1
x 1 dx
x x x
+
+
( ).
ln
17.
3
2
3
6
x dx
x
π
π
cos .
sin
18.
4
2
0
tgx dx
x
π
.
cos
19.
1x x
x x
0
e e
e e
−
−
−
+
dx
20.
1x
x x
0
e dx
e e
−
+
.
21.
2
2
1
dx
4x 8x+
22.
3
x x
0
dx
e e
−
+
ln
.
22.
2
0
dx
1 x
π
+
sin
23.
dx
xx
∫
+
2
1
32
11
24.
∫
−
++
1
1
2)12( dxxx
25.
∫−−
2
0
3)
3
2
2( dxxx
26.
∫
−
−
2
2
)3( dxxx
27.
∫
−
−
4
3
2)4( dxx
29.
∫−
2
1
3
22dx
x
xx
30.
∫e
e
x
dx
1
1
31.
∫
16
1
.dxx
32.
dx
x
xx
e
∫−+
2
1
752
33.
dx
x
x
∫
−
8
132
3
1
4
II. PH NG PHÁP Đ T N PH :ƯƠ Ặ Ẩ Ụ
1.
2
3 2
3
sin xcos xdx
π
π
2.
2
2 3
3
sin xcos xdx
π
π
3.
2
0
sin
1 3
xdx
cosx
π
+
4.
4
0
tgxdx
π
5.
4
6
cot gxdx
π
π
6.
1
2
0
1x x dx+
2
Ths Đ ng Thanh C u-01696900100 ặ ầ BÀI T P NGUYÊN HÀM TÍCH PHÂNẬ
7.
1
2
0
1x x dx−
8.
1
3 2
0
1x x dx+
9.
12
3
0
1
xdx
x+
10.
1
3 2
0
1x x dx−
11.
2
3
1
1
1
dx
x x +
12.
1
2
0
1
1dx
x+
13.
1
2
1
1
2 2dx
x x
−
+ +
14.
1
2
0
1
1
dx
x+
15.
1
2 2
0
1
(1 3 ) dx
x+
16.
2
sin
4
x
e cosxdx
π
π
17.
2
3 2
3
sin xcos xdx
π
π
18.
2
1
2
0
x
e xdx
+
19.
2
3 2
3
sin xcos xdx
π
π
20.
2
sin
4
x
e cosxdx
π
π
21.
2
4
sin
cosx
e xdx
π
π
22.
2
1
2
0
x
e xdx
+
23.
5
0
sin xdx
π
24.
2
2 3
3
sin xcos xdx
π
π
25.
2
0
sin
1 3
xdx
cosx
π
+
32.
12
3
0
1
xdx
x+
33.
1
3 2
0
1x x dx−
34.
2
3
1
1
1
dx
x x +
35.
1
1 ln
e
xdx
x
+
36.
1
sin(ln )
e
xdx
x
37.
1
1 3ln ln
e
x x dx
x
+
38.
2ln 1
1
ex
edx
x
+
39.
2
2
1 ln
ln
e
e
xdx
x x
+
41.
2
1
1 1
xdx
x+ −
42.
1
0
2 1
xdx
x+
43.
1
0
1x x dx+
44.
1
0
1
1dx
x x+ +
45.
1
0
1
1dx
x x+ −
46.
3
1
1xdx
x
+
47.
1
sin(ln )
e
xdx
x
49.
2ln 1
1
ex
edx
x
+
51.
2
2
1
(1 ln )
e
e
dx
xcos x+
53.
( )
2
4
0
sin 1 cosx xdx
π
+
55.
4
2
0
4x dx−
56.
1
2
0
1
dx
x+
57.
dxe x
∫
−
+
0
1
32
58.
∫−
1
0
dxe x
59.
1
3
0
xdx
(2x 1)+
∫
60.
1
0
xdx
2x 1+
∫
61.
1
0
x 1 xdx−
∫
62.
1
2
0
4 11
5 6
xdx
x x
+
+ +
63.
1
2
0
2x 5 dx
x 4x 4
−
− +
∫
64.
33
2
0
xdx
x 2x 1+ +
∫
65.
∫−
2
0sin25
cos
π
dx
x
x
66.
6
6 6
0
(sin cos )x x dx
π
+
67.
3
2
0
4sin
1 cos
xdx
x
π
+
68.
4
2
0
1 sin 2
cos
xdx
x
π
+
69.
2
4
0
cos 2xdx
π
70.
∫+
4
02sin21
2cos
π
dx
x
x
71.
2
6
1 sin 2 cos 2
sin cos
x xdx
x x
π
π
+ +
+
72.
1
0
1
1
x
dx
e+
73.
4
0
1dx
cosx
π
∫
74.
∫+
2
013cos2
3sin
π
dx
x
x
75.
2
5
0
cos xdx
π
76.
∫
−−+
+
0
2
232
22
dx
xx
x
77.
∫++
−
1
1252xx
dx
78.
4
4
0
1
cos dx
x
π
. 79.
4
2
0
sin 4
1 cos
xdx
x
π
+
80.
1
3 2
0
1x x dx−
81.
2
2 3
0
sin 2 (1 sin )x x dx
π
+
82.
2
3 2
0
cos sinx xdx
π
84.
dxxx )sin(cos
4
0
44
∫−
π
86.
15 3 6
0
x (1 x ) dx−
∫
3
Ths Đ ng Thanh C u-01696900100 ặ ầ BÀI T P NGUYÊN HÀM TÍCH PHÂNẬ
87.
6
2
0
cos
6 5sin sin
xdx
x x
π
− +
88.
∫+
2
022 sin4cos
2sin
π
dx
xx
x
89.
4
0
cos sin
3 sin 2
x x dx
x
π
+
+
90.
∫+
2
02
)sin2(
2sin
π
dx
x
x
91.
∫
3
4
2sin
)ln(
π
π
dx
x
tgx
92.
34
0
cos 2
tg x dx
x
93.
∫−+ −
5ln
3ln 32 xx ee
dx
94.
∫−
4
0
8)1(
π
dxxtg
95.
∫+
−
2
4
2sin1
cossin
π
π
dx
x
xx
96.
∫+
+
2
0cos31
sin2sin
π
dx
x
xx
97.
∫+
2
0cos1
cos2sin
π
dx
x
xx
98.
∫+
−
4
0
2
2sin1
sin21
π
dx
x
x
99.
∫−+
2
111 dx
x
x
100.
12
0
1 x dx−
∫
101.
∫+
2
0
sin cos)cos(
π
xdxxe x
102.
∫+
edx
x
xx
1
lnln31
103.
1
2
0
1dx
1 x+
∫
104.
1
2
0
1dx
4 x−
∫
105.
1
2
0
1dx
x x 1− +
∫
106.
1
4 2
0
xdx
x x 1+ +
∫
107.
2
0
1
1 cos sin dx
x x
π
+ +
∫
108.
2
2
2
2
0
xdx
1 x−
∫
109.
22 2
1
x 4 x dx−
∫
110.
2
3
2
2
1dx
x x 1−
∫
101.
32
2
1
9 3x dx
x
+
∫
112.
1
5
0
1
(1 )
xdx
x
−
+
∫
113.
2
2
2
3
1
1dx
x x −
∫
114.
2
0
cos
7 cos2
xdx
x
π
+
∫
115.
14
6
0
1
1
xdx
x
+
+
∫
116.
2
0
cos
1 cos
xdx
x
π
+
∫
117.
∫++
−
0
1222xx
dx
118.
∫++
1
0311 x
dx
119.
∫−
−
2
15
1dx
x
xx
120.
8
2
3
1
1dx
x x +
∫
121.
73
3 2
0
1
xdx
x+
∫
122.
35 2
0
1x x dx+
∫
123.
ln2
x
0
1dx
e 2+
∫
124.
7
3
3
0
1
3 1
xdx
x
+
+
∫
125.
22 3
0
1x x dx+
∫
126.
∫+
32
524xx
dx
II. PH NG PHÁP TÍCH PHÂN T NG PH N:ƯƠ Ừ Ầ
Công th c tích phân t ng ph n : ứ ừ ầ
u( )v'(x) x ( ) ( ) ( ) '( )
b b
b
a
a a
x d u x v x v x u x dx= −
� �
Tich phân cac ham sô dê phat hiên u va dv
@ Dang 1
sin
( )
ax
ax
f x cosax dx
e
β
α
� �
� �
� �
� �
� �
( ) '( )
sin sin
cos
ax ax
u f x du f x dx
ax ax
dv ax dx v cosax dx
e e
= =
� �
� �
� � � �
� �
� �
� � � �
= =
� �
� � � �
� �
� � � �
� � � �
� �
Vi du# $ 1: tinh cac tich phân sau# # #
a/
12
2
0
( 1)
x
x e dx
x+
đăt $
2
2
( 1)
x
u x e
dx
dv x
=
=
+
b/
38
4 3
2
( 1)
x dx
x−
đăt $
5
3
4 3
( 1)
u x
x dx
dv x
=
=
−
c/
1 1 1 1
2 2 2
1 2
2 2 2 2 2 2 2
0 0 0 0
1
(1 ) (1 ) 1 (1 )
dx x x dx x dx
dx I I
x x x x
+ −
= = − = −
+ + + +
� � � �
Tinh I#1
1
2
0
1
dx
x
=+
băng ph ng phap đôi biên sô% ươ # ( # #
4
@ Dang 2:
( ) ln( )f x ax dx
β
α
Đăt$
ln( )
( ) ( )
dx
du
u ax x
dv f x dx v f x dx
=
=
� �
=
=
@ Dang 3:
sin
.� �
� �
� �
ax
ax
e dx
cosax
β
α
Ths Đ ng Thanh C u-01696900100 ặ ầ BÀI T P NGUYÊN HÀM TÍCH PHÂNẬ
Tinh I#2 =
12
2 2
0
(1 )
x dx
x+
băng ph ng phap t ng phân : đăt % ươ # ư% % $
2 2
(1 )
u x
x
dv dx
x
=
=
+
Bài t pậ
1.
3
3
1
ln
e
xdx
x
2.
1
ln
e
x xdx
3.
1
2
0
ln( 1)x x dx
+
4.
2
1
ln
e
x xdx
5.
3
3
1
ln
e
xdx
x
6.
1
ln
e
x xdx
7.
1
2
0
ln( 1)x x dx
+
8.
2
1
ln
e
x xdx
9.
2
0
( osx)s inxx c dx
π
+
10.
1
1
( ) ln
e
x xdx
x
+
11.
2
5
1
ln xdx
x
12.
3
2
0
tanx xdx
π
13.
2
2
1
ln( )x x dx
+
14.
2
0
cosx xdx
π
15.
1
0
x
xe dx
16.
2
0
cos
x
e xdx
π
Tính các tích phân sau
1)
∫
1
0
3
.dxex x
2)
∫−
2
0
cos)1(
π
xdxx
3)
∫−
6
0
3sin)2(
π
xdxx
4)
∫
2
0
2sin.
π
xdxx
5.
∫
e
xdxx
1
ln
6.
∫−
e
dxxx
1
2.ln).1(
7.
∫
3
1
.ln.4 dxxx
8.
1
2
0
ln(3 ).x x dx+
9.
2
2
1
( 1)
x
x e dx+
10.
∫
π
0
.cos. dxxx
11.
∫
2
0
2.cos.
π
dxxx
12.
2
0
sin xdx
π
∫
13.
2
5
1
lnxdx
x
∫
14.
22
0
xcos xdx
π
∫
15.
1x
0
e sinxdx
∫
16)
2
2
0
( 2 )sinx x xdx
π
+
17.
e2
1
xln xdx
∫
18.
3
2
0
x sinxdx
cos x
π
+
∫
19.
2
0
xsinxcos xdx
π
∫
20.
42
0
x(2cos x 1)dx
π
−
∫
21.
∫+
2
0
3sin)cos(
π
xdxxx
22.
12 2x
0
(x 1) e dx+
∫
23.
e2
1
(xlnx) dx
∫
24.
∫−
1
0
2
)2( dxex x
25)
2
1
ln
( 1)
e
e
xdx
x+
∫
26.
12
0
tanx xdx
27.
∫+
1
0
2)1ln( dxxx
28.
π
+
/3
0
cosx.ln(1 cosx)dx
29.
∫
edx
x
x
1
ln
30.
2
2
1
ln(1 x)dx
x
+
∫
31.
∫++
2
0
)1ln()72( dxxx
32.
∫−
3
2
2
)ln( dxxx
III. TÍCH PHÂN HÀM H U T :Ữ Ỷ
1.
∫+−
−
5
3
223
12 dx
xx
x
2.
∫++
b
a
dx
bxax ))((
1
3.
∫+
++
1
0
3
1
1dx
x
xx
4.
dx
x
xx
∫
+
++
1
0
2
3
1
1
5.
∫+
1
0
3
2
)13( dx
x
x
6.
∫++
1
0
22 )3()2(
1dx
xx
7.
∫+
−
2
1
2008
2008
)1(
1dx
xx
x
8.
∫−
3
2
22
4
)1( dx
x
x
9.
∫
+
−
1
0
2
32
)1( dx
x
x
n
n
10.
∫
+
2
1
4
)1(
1dx
xx
11.
∫++
−
2
1
24
2
)23(
3dx
xxx
x
12.
∫
−
+−
++−
0
1
2
23
23
9962 dx
xx
xxx
13.
∫+
2
0
2
4
1dx
x
14.
∫+
1
0
4
1dx
x
x
15.
dx
xx
∫+−
2
0
222
1
16.
∫+
1
0
32 )1( dx
x
x
17.
∫+−
4
2
23 2
1dx
xxx
18.
∫+−
++
3
2
3
2
23
333 dx
xx
xx
19.
∫+
−
2
1
4
2
1
1dx
x
x
20.
∫+
1
0
3
1
1dx
x
21.
∫+
+++
1
0
6
456
1
2dx
x
xxx
22.
∫+
−
1
0
2
4
1
2dx
x
x
23.
1
2
0
1
dx
x x+ +
24.
1
2
0
4 11
5 6
xdx
x x
+
+ +
5