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ffií
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f
A
3
∫
odx = c;
∫
dx√
1− x2
= arcsinx + c;
∫
xµdx =
xµ+1
µ + 1
+ c, µ 6= −1;
∫
dx
1 + x2
= arctan x + c;∫
sinxdx = − cos x + c;
∫
axdx =
ax
ln a
+ c;∫
cos xdx = sinx + c;
∫
dx
x
= ln |x|+ c;∫
sec2 xdx = tan x + c;
∫
chxdx = shx + c;∫
csc 2xdx = − cot x + c;
∫
shxdx = chx + c.
*
1 ;
∫ (
3x2 +
4
x
)
dx.
ü
∫ (
3x2 +
4
x
)
dx = 3
∫
x2dx + 4
∫
dx
x
= 3 · x
3
3
+ 4 ln |x|+ c = x3 + 4 ln |x|+ c.
*
2 ;
∫
x2
1 + x2
dx.
ü
∫
x2
1 + x2
dx =
∫ (
1− 1
1 + x2
)
dx
=
∫
dx−
∫
dx
1 + x2
= x− arctan x + c.
*
3 ;
∫
tan2 xdx.
ü
∫
tan2 xdx =
∫
(sec2 x− 1)dx
=
∫
sec2 xdx−
∫
1dx = tanx− x + c.
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1) �
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B
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dF (ϕ(x)) = F ′(ϕ(x))dϕ(x) = F ′(ϕ(x))ϕ′(x)dx,
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ï
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J ∫
F ′(ϕ(x))ϕ′(x)dx = F (ϕ(x)) + c.
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u = ϕ(x)
A
ªffi«ffic
Nffi)
t∫
f(ϕ(x))ϕ′(x)dx = F (ϕ(x)) + c. (1)
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l
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u
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1 ;
∫
sin3 xdx.
ü
∫
sin3 xdx =
∫
sin2 x(− cos x)′dx
= −
∫
(1− cos2 x)(cos x)′dx.
�
u = cosx, t
B
l
Y
1 � ∫
(1− u2)du = u− u
3
3
+ c,
@ffi³
J ∫
sin3 xdx =
cos3 x
3
− cos x + c.
pffirffiö
(1) ë
Ł
∫
f(ϕ(x))ϕ′(x)dx
u=ϕ(x)
=
∫
f(u)du = F (u) + c
u=ϕ(x)
= F (ϕ(x)) + c.
tffi;
ffi
6
Á
=
@�ffffi8³�fi�fl
Ç8)
*
2 ;
∫
lnx
x
dx.
5
ü
∫
lnx
x
dx
u=lnx
=
∫
udu =
1
2
u2 + c
=
1
2
ln2 x + c.
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¢U£
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6
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ffiD
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ϕ(x)
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C
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¸
Kffi
ë
Ł
∫
f(ϕ(x))ϕ′(x)dx =
∫
f(ϕ(x))dϕ(x)
= F (ϕ(x)) + c.
*
3 ;
∫
x2
√
1 + x3dx.
ü
∫
x2
√
1 + x3dx =
1
3
∫ √
1 + x3d(1 + x3)
=
1
3
1
1 + 12
(1 + x3)1+
1
2 + c
=
2
9
(1 + x3)
3
2 + c.
2) ��ffi
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1)
6
Á
=ffiú
Á
»ffiDqö
x = ϕ(t)
ÃffiÞ
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f(x)dx
6
ffi
ffi
D
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∫
f(x)dx =
∫
f(ϕ(t))ϕ′(t)dt.
pffir
f(ϕ(t))ϕ′(t)
A
vffiMffiN
F (t), P�* x = ϕ(t)
A
ú
MffiN
t = ϕ−1(x), t
B
ffi
6
ffi
ffi²ffi@ffiA
∫
f(x)dx =
∫
f(ϕ(t))ϕ′(t)dt = F (t) + c
= F (ϕ−1(x)) + C.
B
ú
MffiN
6
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A
vFMFN
F (t), t f(x) h I k
A
vFMFN
F (ϕ−1(x)).
f
A
∫
f(x)dx =
∫
f(ϕ(t))ϕ′(t)dt
= F (t) + c = F (ϕ−1(x)) + c. (2)
hffi�;
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4
+ |x| ≤ a, ;
∫ √
a2 − x2dx.
ü
�
x = a sin t, |t| ≤ pi2 . t t = arcsin xa ,
>ffi?
∫ √
a2 − x2dx =
∫
a2 cos2 tdt
= a2
∫
1 + cos 2t
2
dt =
a2
2
t +
a2
4
∫
cos 2td2t
=
a2
2
t +
a2
4
sin 2t + c =
a2
2
t +
a2
2
sin t cos t + c
=
a2
2
arcsin
x
a
+
x
2
√
a2 − x2 + c.
*
5 ;
∫ √
3− 2x− x2dx.
ü
∫ √
3− 2x− x2dx =
∫ √
4− (x + 1)2dx = 4
∫ √
1−
(
x + 1
2
)2
d
x + 1
2
= 2 arcsin
x + 1
2
+
x + 1
2
√
3− 2x− x2 + c.
*
6
+
a > 0, ;
∫
dx√
x2 + a2
.
ü
∫
dx√
x2 + a2
x=a sh t
=
∫
(sh t)′
ch t
dt =
∫
dt = t + c1
= sh−1
x
a
+ c1 = ln
(
x
a
+
√
x2
a2
+ 1
)
+ c1
= ln(x +
√
x2 + a2) + c (c = c1 − ln a).
7
*
7 ;
∫
dx√
x + 1
.
ü
∫
dx√
x + 1
x=t2
=
∫
2tdt
t + 1
=
∫
2dt− 2
∫
dt
t + 1
= 2t− 2
∫
d(1 + t)
1 + t
= 2t− 2 ln(1 + t) + c
= 2
√
x− 2 ln(1 +√x) + c.
¬
:
ºU£
òUó
t =
√
x > 0,
UUD
∫
d(1 + t)
1 + t
= ln(1+t)+c,
ÎU
5Uë
Ł
ln |1+t|+c.
h�C
8
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Ú
D
AFÚFÛED�F
x = ϕ(t)
?�G
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6
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*
8 ;
∫ √
x2 − a2dx (a > 0).
ü
�
x = ach t,
B
>
ch t h (−∞, 0] ® [0,+∞) A Õ 6 2�3 ² D�MFÝ J
N
ê
%8&
i8j
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t ≥ 0. ~ ¬ : ch t > 0, 88DqÃ8Ä x = ach t Ý J8³ J√
x2 − a2 h-iffij [a,+∞) k 6ffi lffiffi )qfffiÇ x ≥ a Ú∫ √
x2 − a2dx x=a ch t=
∫
a2sh2tdt = a2
∫
ch 2t− 1
2
dt
= −a
2
2
t +
a2
2
sh tch t + c1
= −a
2
2
ln(x +
√
x2 − a2) + x
2
√
x2 − a2 + c.
Ç
x ≤ −a Ú D'� x = −ach t (t ≥ 0). @ffiA∫ √
x2 − a2dx = −a2
∫
sh2 tdt =
a2
2
t− a
2
2
sh tch t + c1
=
a2
2
ln(−x +
√
x2 − a2) + x
2
√
x2 − a2 + c2
= −a
2
2
[
ln(−x−
√
x2 − a2)
]
+
x
2
√
x2 − a2 + c.
õ�P
»ffiD
@ffiA
∫ √
x2 − a2dx = −a
2
2
ln |x +
√
x2 − a2|+ x
2
√
x2 − a2 + c.
*
9 ;
∫ √
x2 − a2
x
dx, a > 0.
8
ü
�
x = a sec t,
Ç
x ≥ a Ú D 0 ≤ t < pi2 .
º
Ú
∫ √
x2 − a2
x
dx = a
∫
a tan t
a sec t
· sec t tan tdt
= a
∫
(sec2 t− 1)dt = a tan t− at + c
=
√
x2 − a2 − a arccos a
x
+ c.
x ≤ −a Ú D pi2 < t ≤ pi.
º
Ú
√
x2 − a2 = −a tan t, M
∫ √
x2 − a2
x
dx = −a
∫
(sec2 t− 1)dt = −a tan t + at + c1
=
√
x2 − a2 + a arccos a
x
+ c1
=
√
x2 − a2 − a arccos
(
−a
x
)
+ c.
õ�P
»ffiD
@ffiA ∫ √
x2 − a2
x
dx =
√
x2 − a2 − a arccos a|x| + c.
*
10 ;
∫
dx
x
√
a2 + x2
(a > 0).
ü
x > 0
Ú
D
∫
dx
x
√
a2 + x2
=
∫
dx
x2
√
1 +
(a
x
)2 = −1a
∫
1√
1 +
(a
x
)2 dax
= −1
a
ln
a
x
+
√
1 +
a2
x2
+ c
=
1
a
ln
x
a +
√
a2 + x2
+ c.
Ç
x < 0
Ú
D
∫
dx
x
√
a2 + x2
=
∫
d(−x)
(−x)2
√
1 +
(−ax)2
=
1
a
ln
−x
a +
√
a2 + x2
+ c.
õ�P
»ffiD
@ffiA ∫
dx
x
√
a2 + x2
=
1
a
ln
∣∣∣∣ xa +√a2 + x2
∣∣∣∣+ c.
9
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+
u(x)
®
v(x)
¹ffiKffiLffiD
t
A
(u(x)v(x))′ = u′(x)v(x) + u(x)v′(x).
wffix
Dqpffir
u′(x)v(x)
®
u(x)v′(x)
ffi%
A
vffiMffiNffiD
t�R
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}
A
v8M8N8)
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n è
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ù
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v(x)
K L D
u′(x)v(x)
A
v M N D
t u(x)v′(x)
}
A
vffiMffiNffiD
P
A
∫
u(x)v′(x)dx = u(x)v(x) −
∫
u′(x)v(x)dx.
*
1 α 6= −1, ;
∫
xα lnxdx.
ü
xα =
(
xα+1
α + 1
)
′
,
M��
v(x) = x
α+1
α + 1 , u(x) = lnx, t
A
∫
xα lnxdx =
xα+1
α + 1
lnx− 1
α + 1
∫
xα+1(ln x)′dx
=
xα+1
α + 1
lnx− 1
α + 1
∫
xαdx
=
xα+1
α + 1
(
lnx− 1
α + 1
)
+ c.
U
é
DqÇ
α = 0
Ú
D
@ffi³
J
∫
lnxdx = x lnx− x + c.
B
>ffi�Sffiffi
ùffi
}
K8
ë
Ł
∫
u(x)dv(x) = u(x)v(x) −
∫
v(x)du(x).
ffi
h
ffiffiÚ
D
}
Kffi
0�,
“ �
L
”.
*
2 ;
∫
x2exdx.
10
ü
∫
x2exdx =
∫
x2dex = x2ex −
∫
exdx2
= x2ex − 2
∫
xexdx = x2ex − 2
∫
xdex
= x2ex − 2xex + 2
∫
exdx
= (x2 − 2x + 2)ex + c.
h
'
2
Dqpffirffiö
“ �
L
”
6
MffiNffiÄffiŁ
x2,
@ffi³
Ü
ffiffiIffirffi)
*
3 ;
∫
x sinxdx.
ü
∫
x sinxdx = −
∫
xd cos x = −x cos x +
∫
cos xdx
= sinx− x cos x + c.
*
4 ;
∫
arctan xdx.
ü
∫
arctan xdx = x arctan x−
∫
x
1 + x2
dx
= x arctan x− 1
2
∫
d(1 + x2)
1 + x2
= x arctan x− 1
2
ln(1 + x2) + c.
*
5
+
a 6= 0, ; I =
∫
eax cos bxdx
®
J =
∫
eax sin bxdx.
ü
I =
1
a
∫
cos bxdeax =
1
a
eax cos bx +
b
a
J,
J =
1
a
∫
sin bxdeax =
1
a
eax sin bx− b
a
I.
¼
�ffi>
I
®
J
6
ffi
�ffi%�V�W�X
<ffi=
ffiD
@8³
J
I =
∫
eax cos bxdx =
b sin bx + a cos bx
a2 + b2
eax + c,
J =
∫
eax sin bxdx =
a sin bx− b cos bx
a2 + b2
eax + c.
11
*
6 ; In =
∫
lnn xdx, n ≥ 1.
ü
¤ffi¥
A�Y
ffiùffi
In = x ln
n x−
∫
xd lnn x = x lnn x− n
∫
lnn−1 xdx
= x lnn x− nIn−1,
I0 = x + c.
*
7 ; In =
∫
dx
(a2 + x2)n
, n ≥ 1.
ü
In =
x
(a2 + x2)n
−
∫
xd
1
(a2 + x2)n
=
x
(a2 + x2)n
+ 2n
∫
x2
(a2 + x2)n+1
dx
=
x
(a2 + x2)n
+ 2nIn − 2na2In+1.
@ffi³
J
Y
ffiùffi
I1 =
1
a
arctan
x
a
+ c,
In+1 =
1
2na2
· x
(a2 + x2)n
+
2n− 1
2na2
In, (n ≥ 1).
ffi
A
I2 =
∫
dx
(x2 + a2)2
=
x
2a2(a2 + x2)
+
1
2a3
arctan
x
a
+ c.
*
8
+
a > 0, x > 0, ;
∫
arcsin
√
x
x + a
dx.
ü Z
t = arcsin
√
x
x + a , t x = a tan
2 t.
>ffi?
∫
arcsin
√
x
x + a
dx =
∫
tdx = tx−
∫
xdt
= xt− a
∫
tan2 tdt = xt− a
∫
(sec2 t− 1)dt
= xt + at− a tan t + c
= (x + a) arcsin
√
x
x + a
−√ax + c.
12
§3.3 []\����������
* §3.3.1 ^�_�`�affiçffi��b�c�d�e
P (x) = 0
ô�f�gffi
)
+
n ≥ 0, an 6= 0, a0, a1, · · · , an
¹
?
ffiNffiD
t
P (x) = anx
n + an−1x
n−1 + · · ·+ a1x + a0
u
:
%ffi&
n
V
f�gffi
D
n = ∂oP u
:�f�gffiffi6
VffiNffiDq
ô
|
N
?
ô
V
f�gffi
)
nUè
1 ( hjilk�m )
+
P (x)
?
f�gU
D
Q(x)
?
ô�f�g8
D
t
A�n
%
f�g
q(x)
®
r(x) QffiR
P (x) = q(x)Q(x) + r(x), (1)
ê- r(x) = 0
�
∂or < ∂oQ.
o
pffir
P (x) = 0, t
N
q(x) = r(x) = 0
fffiKffi)
pffir
∂oP < ∂oQ, t
N
q(x) = 0, r(x) = P (x)
fffiKffi)
pffir
∂oP ≥ ∂oQ. +
P (x) = anx
n + an−1x
n−1 + · · ·+ a1x + a0 (an 6= 0),
Q(x) = bmx
m + bm−1x
m−1 + · · ·+ b1x + b0 (bm 6= 0, m ≤ n).
>ffi?
P (x)− an
bm
xn−mQ(x) =
(
an−1 − anbm−1
bm
)
xn−1 + · · · = P1(x).
f
A
P (x) =
an
bm
xn−mQ(x) + P1(x),
ê- P1(x) = 0
�
∂oP1 ≤ n− 1.
pffir
P1(x) = 0
�
∂oP1 ≤ m − 1, t
N
r(x) = P1(x)
f
³
J
(1).
pffir
∂oP1 ≥
∂oQ, t
~8K88ö
P1(x) ë
Ł
Q(x) p
%�2
g8
�q
k
%8&EV8NEr
>
∂oP1
6
f�gU
)
k
8
Á
=
DqB
>
∂oPk
?�s
6
D
@
õ
ff8³
J
Pl(x) = 0
�
∂oPl < ∂
oQ.
º
Ú
D
N
r(x) = Pl(x)
@ffi³
J
(1). êffiÂ�t
&
§-¨
Á
=
@ffi?
y ∂oP = n O
�
Nffi^ffiH
u��ffi)
q
§
q(x)
®
r(x)
6
n
%
²
)qpffir
¸ffiA
P (x) = q1(x)Q(x) + r1(x), ∂
or1 < ∂
oQ
�
r1(x) = 0,
13
t
A
(q1(x)− q(x))Q(x) + (r1(x)− r(x)) = 0. (2)
pffir
q1(x)− q(x) 6= 0,
t (q1(x) − q(x))Q(x)
6
V8N ≥ ∂oQ. Î r1(x) − r(x) = 0
�
r1(x) − r(x)
6
V8N
< ∂oQ.
ºffi
(2) v
ñffi
K
Jffi?
ô�f�gffi
)
8
5
A
q1(x) = q(x).
>ffi?
Dq~
5
A
r1(x) = r(x).
B
l
Y
1
6
§ ¨
Á
=
K88$
Ü
D
pFr
P (x)
®
Q(x)
¹
?
�w
N
f�g8
D
t
q(x)
®
r(x) }
?
�w
N
f�gffi
)
Ç
(1) r(x) = 0
Ú
D
u Q(x) $�x P (x).
+
P (x)
?
%U&
n(n ≥ 1) V f�gU DWpffir P (α) = 0, tUu α ? f�gU P (x) 6
%ffi&�yffi)
nUè
2
+
P (x)
?
n(n ≥ 1) V f�gU D t α ? P (x) 6 y 6�z 5 ` Åffiá ?
x− α $�x P (x).
o
+
P (α) = 0.
B
l
Y
1
K
.
D
A
q(x)
®
|
N
r O
P (x) = (x− α)q(x) + r.
¦
x = α
ÃffiÞ
k
@ffi³
J
r = 0.
ú
Á
»ffiD
t
?�{ffi
¨
w
6
)
nffiè
3 ( |�}�~��� ) z
%ffi&
n(n ≥ 1) V f�gffi ¹�� A %ffi&�yffi)
ºffi&
l
Y
¦
h
MffiN�
=
§-¨
)
ú
Ð
�
l
Y
3
@ffi³
J
� +
P (x)
?
%ffi&
n(n ≥ 1) V f�gffi D t�5 A
P (x) = A(x− α1)n1(x− α2)n2 · · · (x− αs)ns , (3)
ê- A 6= 0, α1, · · · , αs
?
Õ
6
ffiN
(
f
P (x)
Õ
6
y
),
Î
n1 + n2 + · · ·+ ns = n.
14
(3)
6
ni(i = 1, 2, · · · , s)
y
αi
6
NffiD
u αi
?
P (x)
6
ni
yffi)/ffiffi%
&
n
V
f�gffi
A
n
&�y
(
?
N
í
e
y
6
&ffiN
).
nffiè
4 �w
N
f�gffiffi6
ffiN�y��8Ł
y Ü
Offi)
o
+
α
?
n(n ≥ 1) V Â�w N f�gffi
P (x) = anx
n + an−1x
n−1 + · · ·+ a1x + a0
6
ffiN�yffi)
t
A
P (α) = anα
n + an−1α
n−1 + · · ·+ a1α + a0 = 0,
>ffi?
P (α) = a¯nα¯
n + a¯n−1α¯
n−1 + · · ·+ a¯1α¯ + a¯0 = 0,
¬
:
a0, a1, · · · , an
¹
?
Â
Nffi)qffi
a¯0 = a0, · · · , a¯n = an.
M
P (α) = anα¯
n + an−1α¯
n−1 + · · ·+ a1α¯ + a0 = P (α¯) = 0.
f
α¯ }
?
P (x)
6
yffi)
pffir
α, α¯
?
�w
N
f�gffi
P (x)
6
%
y
�� ffiN�yffiD
t
B
l
Y
2
K
.
A
P (x) = (x− α)(x − α¯)P1(x) = (x2 − (α + α¯)x + αα¯)P1(x).
B
>
x2 − (α + α¯)x + αα¯ ? Â�w N f�g8 DWffi $ P (x) ³ 6�ŁU P1(x) }
?
Â
w
N
f�gffi
)
¬
Cffi¤ffi¥8~
A
� +
P (x)
?
%ffi&
n(n ≥ 1) V Â�w N f�gffi ) t�5 A
P (x) = A(x− α1)r1 · · · (x− αk)rk(x2 + β1x + γ1)s1 · · · (x2 + βlx + γl)sl , (4)
ê A 6= 0, αi, βj , γj
¹
?
Â
N D
* β2j−4γj < 0, �
Ç
j 6= j′ Ú (βj , γj) 6= (βj′ , γj′) (i =
1, · · · , k, j = 1, · · · , l), Î r1 + · · · + rk + 2s1 + · · ·+ 2sl = n.
êUÂ (4)
@U?
(3),
Ý
Á
ö
yUÐffiÑ
%
y
�� ffiN�y
6
&ffi%�V
�
Łffi%ffi&
ffi
V
x2 + βx + γ,
B
>
ºffi&
ffi
V
A
%
y
�� �yffiD
8�
é
β2 − 4γ < 0.
§3.3.2 Qffiåffiåffiç��
C��ffi
´
6�f�gffi
¹
?
�w
N
f�g8
D
q�
¨
)
15
+
P (x)
®
Q(x)
¹
?
ô�f�gffi
)
t
f(x) =
P (x)
Q(x)
A
Y
MffiN��
M8Nffi)
pffir
∂oP < ∂oQ, tffiu
P (x)
Q(x)
:�
)
B
l
Y
1
K
.
Dqpffir
Q(x) $
x P (x), t
MffiN P (x)
Q(x)
K
Łffi%ffi&
f�g
p
%ffi&
®
D
f
A
P (x)
Q(x)
= q(x) +
r(x)
Q(x)
(∂or < ∂oQ).
B
>
f�gUU6
U�{ffi
[�+
Dffiffi¤ffi¥UÝ
Û�
�
p��ffiŁ��2
®
�
�2
p�
8
6
E/G8)
nffiè
5
+ P (x)
Q(x)
?
%ffi&
D
Q(x)
,
¼
Ł
(4)
6
)
t
n
%�
A
P (x)
Q(x)
=
k∑
i=1
(
A1
x− α + · · ·+
Ar
(x− α)r
)
+
l∑
j=1
(
B1x + C1
x2 + βx + γ
+ · · ·+ Bsx + Cs
(x2 + βx + γ)s
)
. (5)
h��
%ffi&
®
£8D
α
®
r
¹�ffiT
.
¡� i,
f
ø
Ð�ffië
Ł
αi
®
ri.
Î
A1, · · · , Ar
¹�ffiT
.
k� (i).
f
ø
Ð�ffië
Ł
A
(i)
1 , · · · , A(i)ri .
Õ
��ffi
&
®
£
β γ
®
s
¹
�
.
¡� j, B1, · · · , Bs
®
C1, · · · , Cs }
¹��
.
k� (j).
(5)
P (x)
Q(x)
6
Sffiffi
¼
)
¤ffi¥�ffiT
l
Y
5
6
§-¨
)
|
��
l
w
N���
l
(5)
6
w
N
Ai, Bj , Cj ï
) ¦
(5) v
ñ
D
ê
�
@ffi?
Q(x)(
�
Ô
ffi%ffi&ffi
ôffi6
|
N
¬
(
).
Ý
`
(
v�
ð
ñ
(
x
Õ
V�
6
w
NffiD
@
Kffi
³
J
�ffi>
A,B,C
6
%�V�W�X
<ffi=
ffiDq\
Î
¼ Ü A,B,C.
O
�
Sffiffi
�ffiD
@�J
ö
86
8ffi
¼
Ł
�
�2
86
ffi
©
1◦ Ik =
∫
dx
(x− α)k ;
2◦ Jk =
∫
Bx + C
(x2 + βx + γ)k
dx, β2 − 4γ < 0.
16
; Ik
�ffi[�+
)
Î
Jk =
∫
Bx + C
(x2 + βx + γ)k
dx =
∫
Bx + C[(
x + β2
)2
+ 4γ − β
2
4
]k dx,
� 4γ − β2
4 = p
2, x + β2 = t,
f
A
Jk =
∫ B (t− β2
)
+ C
(t2 + p2)k
dt
=
B
2
∫
d(t2 + p2)
(t2 + p2)k
+
∫
C − Bβ2
(t2 + p2)k
dt.
k
ffið�¡
�
%ffi&
ffi�{8
[�+
D
Î
B
3.2.2
6
'
7
K
.
k
ffið�¡
��ffi
&
ffi
Kffi-B
Y
ffiùffiffie
Ü
)
¬
CffiD
1◦, 2◦
�2
MffiN
6
ffi
¹
?
ßffiï
MffiN8)
§3.3.3
*�¢
*
1 ;
ffi
∫
dx
x2 − a2 (a 6= 0).
ü 1
x2 − a2 =
1
(x− a)(x + a) =
1
2a
(
1
x− a − 1x + a
)
,
M
∫
dx
x2 − a =
1
2a
(∫
dx
x− a −
∫
dx
x + a
)
=
1
2a
ln
∣∣∣∣x− ax + a
∣∣∣∣+ c.
£ffi�¤
|
�ffi6
¼
©
1
(u + a)(u + b)
=
1
b− a
(
1
u + a
− 1
u + b
)
, (a 6= b) (6)
®
1
(u + a)(v − a) =
1
u + v
(
1
u + a
+
1
v − a
)
, (u 6= −v) (7)
*
2 ;
∫
dx
x3 + 1
.
ü + 1
x3 + 1
= 1
(x + 1)(x2 − x + 1) =
A
x + 1 +
Bx + C
x2 − x + 1.
>ffi?ffiA
A + B = 0;
B −A + C = 0;
A + C = 1.
17
¼
³
A = 13 , B = −13 , C = 23.
M
∫
dx
x3 + 1
=
1
3
∫
dx
x + 1
− 1
3
∫
x− 2
x2 − x + 1dx
=
1
3
ln |x + 1| − 1
6
∫
2x− 1− 3
x2 − x + 1dx
=
1
6
ln
(x + 1)2
x2 − x + 1 +
1
2
∫
dx
x2 − x + 1
=
1
6
ln
(x + 1)2
x2 − x + 1 +
1
2
∫
dx(
x− 12
)2
+ 34
=
1
6
ln
(x + 1)2
x2 − x + 1 +
1√
3
arctan
2x− 1√
3
+ c.
*
3 ;
∫
x3 + 1
x4 − 3x3 + 3x2 − xdx.
ü
x4 − 3x3 + 3x2 − x = x(x− 1)3. +
x3 + 1
x(x− 1)3 =
A
x
+
B
x− 1 +
C
(x− 1)2 +
D
(x− 1)3 .
>ffi?
x3 + 1 = A(x− 1)3 + Bx(x− 1)2 + Cx(x− 1) + Dx.
(
ñ
x
Õ
V�
6
w
NffiD
³
A + B = 1;
−3A− 2B + C = 0;
3A + B − C + D = 0;
−A = 1.
¼
³
A = −1, B = 2, C = 1, D = 2. M A∫
x3 + 1
x4 − 3x3 + 3x2 − xdx = −
∫
dx
x
+ 2
∫
dx
x− 1 +
∫
dx
(x− 1)2 + 2
∫
dx
(x− 1)3
= ln
(x− 1)2
|x| −
1
(x− 1)2 −
1
x− 1 + c
= ln
(x− 1)2
|x| −
x
(x− 1)2 + c.
O
�
SUU
�
?
;
A
Y
MffiN
6
ffi
6ffi÷Uøffi<
�ffiD¥�¦�¦�§�¨�©�ª
6
í
e
D
ffi
h
K
JffiÚ
Ð
Ç�«�¬
O
�
D
Î����®
�U
é
68<
�8)
18
*
4 ;
∫
x2 + 1
x4 + 1
dx.
ü
x4 + 1 = (x2 −√2x + 1)(x2 +√2x + 1). ¯ � (7) K ³
1
x4 + 1
=
1
2(x2 + 1)
(
1
x2 −√2x + 1 +
1
x2 +
√
2x + 1
)
,
M
∫
x2 + 1
x4 + 1
dx =
1
2
∫
dx
x2 −√2x + 1 +
1
2
∫
dx
x2 +
√
2x + 1
=
1
2
∫ d(x− √22
)
(
x−
√
2
2
)2
+ 12
+
1
2
∫ d(x + √22
)
(
x +
√
2
2
)2
+ 12
=
√
2
2
arctan(
√
2x− 1) +
√
2
2
arctan(
√
2x + 1) + c.
*
5 ;
∫
x2 − 1
x4 + 1
dx.
ü
∫
x2 − 1
x4 + 1
dx =
∫ 1− 1
x2
x2 + 1
x2
dx =
∫ d(x + 1x)(
x + 1x
)2
− 2
=
∫ d(x + 1x)(
x + 1x −
√
2
)(
x + 1x +
√
2
)
=
1
2
√
2
∫ d(x + 1x −√2)
x + 1x −
√
2
− 1
2
√
2
∫ d(x + 1x +√2)
x + 1x +
√
2
=
1
2
√
2
ln
x2 −√2x + 1
x2 +
√
2x + 1
+ c.
kE°
6
MFN
h
6
Ú²±
D²³�´
`
; x 6= 0. º ÚFFFFé h (−∞, 0) ®
(0,+∞) k aUb )µ�¶�� ³ J 6 v8MffiN h x = 0 } ? KUL 6 D >ffi? x 6= 0 6�·�¸ ¶
3
N�¹
)
ffi
ï
h (−∞,+∞) k Ł�Xffi)
*
6 ;
∫
dx
x(1 + x7)
.
19
ü
∫
dx
x(1 + x7)
=
∫
x6dx
x7(1 + x7)
=
1
7
(∫
dx7
x7
−
∫
d(x7 + 1)
x7 + 1
)
=
1
7
ln
x7
x7 + 1
+ c.
*
7 ;
∫
x4 + 1
x6 + 1
dx.
ü
∫
x4 + 1
x6 + 1
dx =
∫ (
x4 − x2 + 1
x6 + 1
+
x2
x6 + 1
)
dx
=
∫
dx
x2 + 1
+
1
3
∫
dx3
(x3)2 + 1
= arctan x +
1
3
arctan(x3) + c.
§3.3.4 º�»�^
è
çffi�ffiäffiå
+
R(u, v)
?E�F>
u, v
6
A
Y
MFNF)qf B
&
¶ ±
u, v
´
ÁFA
·
V½¼
t
2ffie
³
J
6
M8N8D
w8x
R(u, v)
Kffi
ë
Ł
&
ffi
�
f�g886EŁ
)
º8£8¤8¥
R(cos x, sinx)
6
ffi
E/Gffi)
1) ¾
J
ÃffiÄ
�
t = tan x2 ( ¾
J
ÃffiÄ
), t
A
cos x = 2 cos2
x
2
− 1 = 1− t
2
1 + t2
,
sinx = 2 sin
x
2
cos
x
2
= 2 tan
x
2
· cos2 x
2
=
2t
1 + t2
,
dx = 2d arctan t =
2
1 + t2
dt.
>ffi? ∫
R(cos x, sinx)dx = 2
∫
R
(
1− t2
1 + t2
,
2t
1 + t2
)
dt
1 + t2
.
º
?
t
6
A
Y
MffiNffiD
@
Kffi
�
S8ffi
�
8
)
20
*
1 ;
∫
dx
5 + 4 sinx
.
ü
�
t = tan x2 . t
∫
dx
5 + 4 sinx
= 2
∫
dt
5t2 + 8t + 5
=
2
5
∫ d(t + 45
)
(
t + 45
)2
+
(
3
5
)2
=
2
3
arctan
5t + 4
3
+ c.
ê- t = tan x2 .
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tanx
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A
Y
MffiN
R(tan x),
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ffi
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t = tanx
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∫
R(tanx)dx =
∫
R(t)
1 + t2
dt.
*
2 ;
∫
sin2 x cos x
sinx + cos x
dx.
ü
B
sin2 x = 1 − 1
1 + tan2 x
, cos xsinx + cos x =
1
1 + tanx .
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8
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t = tan x. ∫
sin2 x cos x
sinx + cos x
dx =
∫
t2
(1 + t)(1 + t2)2
dt
=
1
4
∫ (
1
1 + t
− t− 1
1 + t2
+
2t− 2
(1 + t2)2
)
dt
=
1
4
ln
|1 + t|√
1 + t2
− 1 + t
4(1 + t2)
+ c
=
1
4
ln | sinx + cos x| − 1
4
cos x(cos x + sinx) + c.
*
3 ;
∫
1− tan x
1 + tan x
dx.
ü
�
t = tan x, t∫
1− tanx
1 + tanx
dx =
∫
1− t
1 + t
· dt
1 + t2
=
∫ (
1
1 + t
− t
1 + t2
)
dt
= ln
|1 + t|√
1 + t2
+ C = ln | cos x + sinx|+ c.
3)
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t = cos x
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t = sinx
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