Ôn tập nguyên hàm và tích phân

Lê Xuân Manh

§T: 0373840709


 !"#
$%&#'()*+#
$,- ./'()*+#
$0 ''()*+# 
$12'3/,45
* ,,456'47'4489:;<+47'44='4!
$>'3/?@<+#A:;'BC'./,45DE'FG
H3I'!+JE
$,'()*+#./#A:;+#:;8' KLMBC',-<+ ''()*+#N19
:;<+'()*+#='4!O
$,,45./#A:;+#:;8' KLMBC',-<+ ''()*+#N19
:;<+,45='4!='4!O
$P4BC',458Q,B&,F4R'
$P4BC',458Q,Q,<SQ
<-8TCQ
UV()!"#W
X%&#'()*+#W
$12'3/W !
"#$%&$%'()
$12YVW*+,) !
a/-(.&$%/0) !
1234.* !564)7!*$%&$%/
XH,- ./'()*+#
$12YVW3"8!95
!2
[ ]

K O K O Z[ \KZOBZ] 'KZOBZf x g x d+
∫ ∫ ∫
12
 \KZOBZ[ \KZOBZ
∫ ∫
'(:;;<
X^0 ''()*+# 

 _BZ[
∫

 BZ[ BZ x C= +
∫ ∫
H
( )


 Z BZ[ 
]
x C
α α
α
α
+
+ ≠ −
∫
^

 BZ[Y Z
Z

C+
∫
`abY+6':;_/c
a/

 :ZBZ[d E:Z]

c
∫
b/

 E:ZBZ[ :Z]

∫
c/
Z Z

 e BZ[ e ]

∫
d/
Z Z

 / BZ[ / ] K_ O
Y/
a< ≠
∫

1<2f'W;'g()5
1

Lê Xuân Manh

§T: 0373840709
h
a/
H

 BZ[/Z]
E: x
∫
b/
H

 BZ[dEZ]
: x
∫
X`,45
X`X12'3/,45WE+#:;\Y*CD*%<+/NY+/:;-F(A%X(i
Y+#A'()*+#.\D*%F&(:;
iKOjiK/O
87k'GY+,45./\=/8<+,&(Y+
K O Z
b
a
f x d
∫
lVW
$(/m/'G
K O Z
b

a
f x d
∫
Y+,45./\D*8EJn/op
$
K O Z
b
a
f x d
∫
=
K O
b
a
F x
= F(b) – F(a)
* ,45f'4C(A<+EY-),45
K O Z
b
a
f x d
∫
=
K O
b
a
f t dt
∫
=
K O

b
a
f v dv
∫
= ....
X`XH,-./,45

K O Z _
a
a
f x d =
∫
H
K O Z K O Z
b a
a b
f x d f x d= −
∫ ∫
^
K O Z K O Z] K O Z
b c b
a a c
f x d f x d f x d=
∫ ∫ ∫
`
n K O K Op Z K O Z] K O Z
b b b
a a a
f x g x d f x d f x d+ =
∫ ∫ ∫

h
K O Z K O Z
b b
a a
kf x d k f x d=
∫ ∫
2/ BJ'EDE'7'
HXX,'()*+#6':MBC',-<+'()*+# 
W,'()*+#./+#:;:/(
/f(x) = x
3
+
^
 H
x x
+
 f(x) =
K OK Ox x x+ − +
f(x) =
^ H
H
x x
x
− −
Bf(x) = x
4
+ 3x
3
-5x + 2
W,'()*+#./+#:;:/(

/f(x) =
H
H E:HZ]H:^Z]Zc \KZO[`:
H
Z
\KZO[
^
q h
 
K OK Ox x
x x
− −
Bf(x) =
H

H: Z]HE:Z]Z
x
+
a,BC^WF#'()*+#iKZO./+#:;
1<2f'W;'g()5
2
Lê Xuân Manh

§T: 0373840709
a/ f(x) =
H
H E:HZ]H:^Z]Zc
biết F(
^
π

)= - 3
b/ f(x) =
^ H
H
x x
x
− −
biết F((-3) = 10
HXH,'()*+#6'47'4489:;
12YVW&$%=49'&$%9!
>$%?@=") !8
A
 \K(OB([ K OF u C+
∫
5
 \n(KZOpBZ[ n K OpF u x C+
∫
W,'()*+#./+#:;:/(
/
E:Z
:e xdx
∫

h
Kh ^Ox dx+
∫

^
H
E:

tg x
dx
x
∫
B
:Z
E:e xdx
∫
W,
/
H
^
r
dZ
x
dx
∫

H
`
x x dx−
∫

H

K O
dx
x x+
∫
B



x
dx
e +
∫
W,
/
h
Z
: E:
^ ^
x
c dx
∫

^ H
`
x x dx−
∫

H


dx
x x−
∫
B
HY ^x
dx

x
+
∫
W,
/
 /Zdx
∫

E Zdx
∫

^
K/  /ZOdx+
∫
B
^
KE Z]E ZOdx
∫
g/
H

s r
dx
x x− +
∫
h/
H
^ H
s r
x

dx
x x
+
+ +
∫
HXX^,'()*+#6'47'44='4!
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 (KZO<tKZOBZ[ K O K O K O tK Ou x v x v x u x du−
∫ ∫
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/\KZO[Z
H
:HZ\KZO[Z
H
E:Z
\KZO[Z
H
e
Z
B\KZO[Z
^
YKHZO
W,
a/
:Zx dx
∫
b/
Z
K Ox e dx+
∫

c/
KH O:x xdx+
∫
(x) d/
`
: E:x xdx
∫
e/
h
Kh ^Ox dx+
∫
g/
Z
e :Zdx
∫
h/
Yx xdx
∫
k/
H
Yx xdx
∫
W,
a/
H
:Zx dx
∫
b/
dZ
xe dx

∫
c/
:
^
x
x dx
∫
(x) d/
: H
x
e xdx
∫
1<2f'W;'g()5
3
Lê Xuân Manh

§T: 0373840709
,
a/
H x
e dx
+
∫
b/
H
/x xdx
∫
c/
E:KYZOc dx
∫

(x) d/
H
YK Ox x dx+
∫
HXHX,,456':MBC',-<+'()*+# 
W,
a/

^
_
K Ox x dx+ +
∫
b/
H
H

 
K O
e
x x dx
x x
+ + +
∫

c/
^

Hx dx−
∫
d/

H

x dx+
∫

W,
a/
H
^
KH: ^ Ox cosx x dx
π
π
+ +
∫
b/

_
K O
x
e x dx+
∫

c/

^
_
K Ox x x dx+
∫
d/
H


K OK Ox x x dx+ − +
∫

W,
a/
H
^

K^: H Ox cosx dx
x
π
π
+ +
∫
b/

H
_
K O
x
e x dx+ +
∫

c/
H
H
^

K Ox x x x dx+ +

∫
d/
H

K OK Ox x x dx− + +
∫

HXHXH,,456'47'4489:;
f'B9:;
K O
K O
n(K Op(tKZO Z K O
u b
b
a u a
f x d f u du=
∫ ∫
W,
a/
H
^ H
^
: xcos xdx
π
π
∫
b/
H
H ^
^

: xcos xdx
π
π
∫

c/
H
_
:
 ^
x
dx
cosx
π
+
∫
d/
`
_
tgxdx
π
∫

g/
`
s
E gxdx
π
π
∫

h/
s
_
 `: xcosxdx
π
+
∫
W,
a/

H
_
x x dx+
∫
b/

H
_
x x dx−
∫

c/

^ H
_
x x dx+
∫
d/

H

^
_

x
dx
x +
∫

1<2f'W;'g()5
4
Lê Xuân Manh

§T: 0373840709
g/

^ H
_
x x dx−
∫
h/
H
^



dx
x x +
∫
W,
a/


H
_


dx
x+
∫
b/

H


H H
dx
x x
−
+ +
∫

c/

H
_


dx
x +
∫
d/


H H
_

K ^ O
dx
x+
∫
g/
H
:
`
x
e cosxdx
π
π
∫
h/
H
`
:
cosx
e xdx
π
π
∫

f/
H


H
_
x
e xdx
+
∫
k/
H
^ H
^
: xcos xdx
π
π
∫

W,
1/
H
:
`
x
e cosxdx
π
π
∫
2/
H
`
:
cosx

e xdx
π
π
∫
3/
H

H
_
x
e xdx
+
∫
W,
1/
H
^ H
^
: xcos xdx
π
π
∫
2/
H
H ^
^
: xcos xdx
π
π
∫

3/
H
_
:
 ^
x
dx
cosx
π
+
∫

4/
`
_
tgxdx
π
∫
5/
`
s
E gxdx
π
π
∫
6/
s
_
 `: xcosxdx
π

+
∫
W,
1/

H
_
x x dx+
∫
2/

H
_
x x dx−
∫
3/

^ H
_
x x dx+
∫
4/

H
^
_

x
dx
x +

∫
5/

^ H
_
x x dx−
∫
6/
H
^



dx
x x +
∫
W,
1/

 Y
e
x
dx
x
+
∫
2/

:KY O
e

x
dx
x
∫
3/

 ^Y Y
e
x x
dx
x
+
∫
4/
HY 

e
x
e
dx
x
+
∫
5/
H
H
 Y
Y
e
e

x
dx
x x
+
∫
6/
H
H

K Y O
e
e
dx
cos x+
∫
W,
1/
H

 
x
dx
x+ −
∫
2/

_
H 
x
dx

x +
∫
3/

_
x x dx+
∫
1<2f'W;'g()5
5