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S�`i AA
6BM�H
RyR
5AMi2;`�iBQM
8XR AM/2}MBi2 AMi2;`�H
8XRXR �MiB/2`Bp�iBp2
/27BMBiBQM � 7mM+iBQM F Bb +�HH2/ �M �MiB/2`Bp�iBp2 Q7 � 7mM+iBQM f QM � ;Bp2M QT2M
BMi2`p�H B7 F ′(x) = f(x) 7Q` �HH x BM i?2 BMi2`p�HX
h?2 T`Q+2bb Q7 }M/BM; �MiB/2`Bp�iBp2b Bb +�HH2/ �MiB/Bz2`2MiB�iBQM Q` BMi2;`�iBQMX h?mb- B7
d
dx[F (x)] = f(x) U8XRV
i?2M BMi2;`�iBM; UQ` �MiB/Bz2`2MiB�iBM;V i?2 7mM+iBQM f(x) T`Q/m+2b �M �MiB/2`Bp�iBp2 Q7 i?2 7Q`K
F (x) + C �b BM i?2 7QHHQrBM; h?2Q`2KX
h?2Q`2K 8XR A7 F (x) Bb �Mv �MiB/2`Bp�iBp2 Q7 f(x) QM �M QT2M BMi2`p�H- i?2M 7Q`
�Mv +QMbi�Mi C i?2 7mM+iBQM F (x) + C Bb �HbQ �M �MiB/2`Bp�iBp2 QM i?�i BMi2`p�HX
JQ`2Qp2`- 2�+? �MiB/2`Bp�iBp2 Q7 f(x) QM i?2 BMi2`p�H +�M #2 2tT`2bb2/ BM i?2 7Q`K
F (x) + C #v +?QQbBM; i?2 +QMbi�Mi C �TT`QT`B�i2HvX
hQ 2KT?�bBx2 i?Bb T`Q+2bb- 1[m�iBQM U8XRV Bb `2+�bi mbBM; BMi2;`�H MQi�iBQM-
∫f(x)dx = F (x) + C U8XkV
r?2`2 C Bb mM/2`biQQ/ iQ `2T`2b2Mi �M �`#Bi`�`v +QMbi�MiX LQiB+2 i?�i i?2 p�Hm2b Q7 C `2bmHi iQ i?2
b?B7iBM; Q7 i?2 7mM+iBQM F (x) mT Q` /QrMX
Ryk
Ryj
6Q` 2t�KTH2- 1
3x3,
1
3x3 + 1,
1
3x3 − 3,
1
3x3 −
√2 �`2 �HH �MiB/2`Bp�iBp2b Q7 f(x) = . . . . . . . . . X
8XRXk AMi2;`�iBQM 6Q`KmH�b
.Bz2`2MiB�iBQM 6Q`KmH� AMi2;`�iBQM 6Q`KmH�
RX d
dx(C) = 0 RX
∫0dx = C
kX d
dx[kx] = k kX
∫kdx = kx+ C
jX d
dx[kf(x)] = kf ′(x) jX
∫[kf(x)]dx = k
∫f(x)dx
9X d
dx[f(x)± g(x)] = f ′(x)± g′(x) 9X
∫[f(x)± g(x)]dx =
∫f(x)dx±
∫g(x)dx
8X d
dx[xn] = nxn−1 8X
∫xndx =
xn+1
n+ 1+ C, n #= −1
eX d
dx[ln |x|] = 1
xeX
∫1
xdx = ln |x|+ C
dX d
dx[ex] = ex dX
∫exdx = ex + C
3X d
dx[ax] = ax ln a, a > 0 �M/ a #= 1 3X
∫axdx =
ax
ln a+ C, a > 0 �M/ a #= 1
NX d
dx[sinx] = cosx NX
∫cosxdx = sinx+ C
RyX d
dx[cosx] = − sinx RyX
∫sinxdx = − cosx+ C
RRX d
dx[tanx] = sec2 x RRX
∫sec2 xdx = tanx+ C
RkX d
dx[cotx] = −cosec2x RkX
∫cosec2xdx = − cotx+ C
RjX d
dx[secx] = secx tanx RjX
∫secx tanxdx = secx+ C
R9X d
dx[cosecx] = −cosecx cotx R9X
∫cosecx cotxdx = −cosecx+ C
R8X d
dx[arcsinx] =
1√1− x2
R8X∫
1√1− x2
dx = arcsinx+ C
ReX d
dx[arctanx] =
1
1 + x2ReX
∫1
1 + x2dx = arctanx+ C
RdX d
dx[ln | secx|] = tanx RdX
∫tanxdx = ln | secx|+ C
R3X d
dx[ln | sinx|] = cotx R3X
∫cotxdx = ln | sinx|+ C
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Ry9
1t�KTH2 8XR
U�V∫(x6 − 7x+ 4)dx =
U#V∫
x5 + 2x3 − 1
x4dx =
U+V∫(√x+
13√x)dx =
U/V∫(ex + 2x)dx =
U2V∫(4 sinx+ 2 cosx)dx =
U7V (3√
1− x2− 2
1 + x2)dx =
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Ry8
8Xk AMi2;`�iBQM #v am#biBimiBQM
h?2Q`2K 8Xk G2i u #2 � 7mM+iBQM Q7 x �M/ f #2 � 7mM+iBQM Q7 uX h?2M
∫[f(u)]du =
∫[f(u(x))u′(x)]dx.
1t�KTH2 8Xk 1p�Hm�i2∫(2x+ 1)100dx
1t�KTH2 8Xj 1p�Hm�i2∫
x2
(3x3 − 2)9dx
1t�KTH2 8X9 1p�Hm�i2∫
cos(5x)dx
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rye
1t�KTH2 8X8 1p�Hm�i2∫ cos(
√x)√
xdx
1t�KTH2 8Xe 1p�Hm�i2∫
ex sec2(ex + 1)dx
1t�KTH2 8Xd 1p�Hm�i2∫
1√2− x2
dx
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
1t2`+Bb2 8�
1p�Hm�i2 i?2 BMi2;`�HbX
RX∫
x1/8dx
kX∫
1
x6dx
jX∫
3x(2x− 7)dx
9X∫
x+√7dx
8X∫(x
3+ 3x)dx
eX∫(x3 + 1)
√xdx
dX∫
x4 + 7x3 − 5x2 + 1
x2dx
3X∫
x5 −√x
x3dx
NX∫(6ex − lnx)dx
RyX∫(1 + sinx)dx
RRX∫(5 sec2 x+ cosec2x)dx
RkX∫
sin 2x
cosxdx
RjX∫
secx
sec2 x− 1dx
R9X∫(1 + sinx+ 8 cosx)dx
R8X∫(
15√1− x2
− 21
1 + x2)dx
ReX∫
x√4− x2dx
RdX∫
x4√x5 − 9
dx
R3X∫
ex
1 + e2xdx
RNX∫
1√16− x2
dx
kyX∫
sec2(3x)dx
kRX∫
cosx
2− sinxdx
kkX∫
tan2 x+ 1
cotxdx
kjX∫
7lnx
xdx
Ryd
6h2+?MB[m2b Q7 AMi2;`�iBQM
eXR Pp2`pB2r Q7 AMi2;`�iBQM J2i?Q/b
� `2pB2r Q7 7�KBHB�` BMi2;`�iBQM 7Q`KmH�b
RX∫
du = u+ C
kX∫
undu =un+1
n+ 1+ C, n #= −1
jX∫
1
udu = ln |u|+ C
9X∫
au du =au
ln a+ C, a > 0, a #= 1
8X∫
eu du = eu + C
eX∫
sinu du = − cosu+ C
dX∫
cosu du = sinu+ C
3X∫
sec2u du = tanu+ C
NX∫
csc2u du = − cotu+ C
RyX∫
sec u tanu du = secu+ C
RRX∫
csc u cotu du = − cscu+ C
RkX∫
tanu du = ln| secu|+ C
RjX∫
cotu du = ln| sinu|+ C
R9X∫
du√a2 − u2
= arcsin(u
a) + C
R8X∫
du
a2 + u2=
1
aarctan(
u
a) + C
Ry3
RyN
eXk AMi2;`�iBQM #v S�`ib
eXkXR h?2 S`Q/m+i _mH2 pb AMi2;`�iBQM #v S�`ib
G2i G(x) #2 �Mv �MiB/2`Bp�iBp2 Q7 g(x) c G′(x) = g(x)
d
dx[f(x)G(x)] = f(x)G′(x) + f ′(x)G(x) = f(x)g(x) + f ′(x)G(x)
∫[f(x)g(x) + f ′(x)G(x)] dx = f(x)G(x)
∫f(x)g(x) dx = f(x)G(x)−
∫f ′(x)G(x) dx
G2i u = f(x), du = f ′(x)dx �M/ v = G(x), dv = g(x) dx
∫udv = uv −
∫vdu
1t�KTH2 eXR 1p�Hm�i2∫
x bBM x dxX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
RRy
1t�KTH2 eXk 1p�Hm�i2∫
x3HM x dxX
eXkXk _2T2�i2/ AMi2;`�iBQM #v S�`ib
1t�KTH2 eXj 1p�Hm�i2∫
x2ex dxX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
RRR
1t�KTH2 eX9 1p�Hm�i2∫
ex cosx dxX
1t�KTH2 eX8 1p�Hm�i2∫
arctanx dxX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
1t2`+Bb2 e�
1p�Hm�i2 i?2 7QHHQrBM; BMi2;`�HbX
RX∫
x sinx
2dx
kX∫ √
x lnx dx
jX∫
xsec2x dx
9X∫
(lnx)2 dx
8X∫x2 sinx dx
eX∫
xcos2x dx
dX∫
e√3x+9 dx
3X∫
sin(lnx) dx
NX∫
xe3x dx
RyX∫
xe−2x dx
RRX∫
x2ex dx
RkX∫
x2e−2x dx
RjX∫
x sin 3x dx
R9X∫
x cos 2x dx
R8X∫
x2 cosx dx
ReX∫
x lnx dx
RdX∫
ln(3x− 2) dx
R3X∫
ln(x2 + 4) dx
RNX∫
arcsinx dx
kyX∫
arccos(2x) dx
kRX∫
arctan(3x) dx
kkX∫
x arctanx dx
kjX∫
ex sinx dx
k9X∫
e3x cos(2x) dx
k8X∫
cos(lnx) dx
keX∫
x tan2 x dx
kdX∫
x3ex2dx
k3X∫
lnx√x
dx
kNX∫
xex
(x+ 1)2dx
jyX∫ π
0(x+ x cosx) dx
jRX∫ 2
0xe2x dx
jkX∫ 1
0xe−5x dx
jjX∫ e
1x2 lnx dx
j9X∫ e
√e
lnx
x2dx
j8X∫ 1
−1ln(x+ 2) dx
jeX∫ √
3/2
0arcsinx dx
jdX∫ 4
2sec−1√x dx
j3X∫ 2
1x sec−1 x dx
jNX∫ π
0x sin 2x dx
9yX∫ 3
1
√x arctan
√x dx
9RX∫ 2
0ln(x2 + 1) dx
RRk
RRj
eXj AMi2;`�iBM; h`B;QMQK2i`B+ 6mM+iBQMb
q2 bi�`i i?2 b2+iBQM #v `2pB2rBM; BKTQ`i�Mi i`B;QMQK2i`B+ B/2MiBiB2b �b 7QHHQrBM;,
sin2 x+ cos2 x = 1 tan2 x = sec2 x− 1
sin 2x = 2 sinx cosx cos 2x = cos2 x− sin2 x
sin2 x = 12(1− cos 2x) cos2 x = 1
2(1 + cos 2x)
eXjXR AMi2;`�iBM; S`Q/m+ib Q7 aBM2b �M/ *QbBM2b
A7 m �M/ n �`2 TQbBiBp2 BMi2;2`b- i?2 BMi2;`�H∫
sinmx cosnx dx +�M #2 2p�Hm�i2/ #v QM2 Q7 i?2
7QHHQrBM; T`Q+2/m`2b- /2T2M/BM; QM r?2i?2` m �M/ n �`2 2p2M Q` Q//X∫
sinmx cosnx dx S`Q+2/m`2 `2H2p�Mi B/2MiBiv
m Q// b2i sinm x = sinm−1x sinx sin2 x = 1− cos2 x
n Q// b2i cosnx = cosn−1x cosx cos2x = 1− sin2x
m �M/ n 2p2M b2i sin2x = 12(1− cos 2x) cos2x = 1
2(1 + cos 2x)
Q` b2i cos2x = 12(1 + cos 2x) sin2x = 1
2(1− cos 2x)
1t�KTH2 eXe 1p�Hm�i2∫
sin3x dxX
1t�KTH2 eXd 1p�Hm�i2∫
cos5x dxX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
RR9
1t�KTH2 eX3 1p�Hm�i2∫
sin2x cos3x dxX
1t�KTH2 eXN 1p�Hm�i2∫
cos1/3 x sin3 x dxX
1t�KTH2 eXRy 1p�Hm�i2∫ (
1 + sinx)2
dxX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
RR8
eXjXk AMi2;`�iBM; S`Q/m+ib Q7 aBM2b �M/ *QbBM2b rBi? .Bz2`2Mi �M;H2b
AMi2;`�Hb Q7 i?2 7Q`K
∫sinmx cosnx dx,
∫sinmx sinnx dx,
∫cosmx cosnx dx
+�M #2 2p�Hm�i2/ mbBM; i?2 7QHHQrBM; B/2MiBiB2b,
sin(mx) cos(nx) = 12
{sin(m+ n)x+ sin(m− n)x
}
sin(mx) sin(nx) = 12
{cos(m− n)x− cos(m+ n)x
}
cos(mx) cos(nx) = 12
{cos(m+ n)x+ cos(m− n)x
}
1t�KTH2 eXRR 1p�Hm�i2∫
sin 3x cos 5x dxX
1t�KTH2 eXRk 1p�Hm�i2∫
sin 3x sin5x
2dxX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
1t2`+Bb2 e#
1p�Hm�i2 i?2 7QHHQrBM; BMi2;`�HbX
RX∫
sin3x cos2x dx
kX∫
cos3x
sinxdx
jX∫
cos5/3x sinx dx
9X∫
sin4x dx
8X∫
cos4x sin4x dx
eX∫
cos3 x sinx dx
dX∫
sin5 3x cos 3x dx
3X∫
sin2 5x dx
NX∫
cos2 3x dx
RyX∫
sin3 ax dx
RRX∫
cos3 ax dx
RkX∫
sinx cos3 x dx
RjX∫
sin2 x cos2 x dx
R9X∫
sin2 x cos4 x dx
R8X∫
sin 2x cos 3x dx
ReX∫
sin 3x cos 2x dx
RdX∫
sinx cos(x/2) dx
R3X∫
sin 7x sin 2x dx
RNX∫
cos 4x cos 9x dx
RRe
RRd
eX9 h`B;QMQK2i`B+ am#biBimiBQM
q2 rBHH #2 +QM+2`M2/ rBi? BMi2;`�Hb i?�i +QMi�BM 2tT`2bbBQMb Q7 i?2 7Q`K√
a2 − u2,√u2 ± a2X
√a2 − u2 u = a sin θ −π
2< θ <
π
2√a2 + u2 u = a tan θ −π
2< θ <
π
2√u2 − a2 u = a sec θ 0 ≤ θ <
π
2,π
2< θ ≤ π
1t�KTH2 eXRj 1p�Hm�i2∫
dx√9 + x2
X
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
RR3
1t�KTH2 eXR9 1p�Hm�i2∫
x3√9− x2
dxX
1t�KTH2 eXR8 1p�Hm�i2∫
dx
x2√4x2 − 3
X
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
1t2`+Bb2 e+
1p�Hm�i2 i?2 7QHHQrBM; BMi2;`�HbX
RX∫ √
4− x2 dx
kX∫ √
25− x2 dx
jX∫
dx√4x2 − 49
9X∫
xj√x2 + 4
dx
8X∫
dx
(x2 − 1)3/2
eX∫ √
1− x2
x2dx
dX∫
8
(4x2 + 1)2dx
3X∫
dx√x2 + 2x− 3
NX∫
dx
(x2 − 2x+ 10)32
RyX∫
x+ e√4x− x2
dx
RRX∫
dx
x2√x2 − 16
RkX∫
3x3√1− x2
dx
RjX∫
x2√16− x2
dx
R9X∫ √
x2 − 9
xdx
R8X∫
3x3√x2 − 25
dx
ReX∫
cosx√2− sin2 x
dx
RdX∫ √
2x2 − 4
xdx
R3X∫
x3
(3 + x2)5/2dx
RNX∫
x2√5 + x2
dx
kyX∫
dx
x2√9− x2
kRX∫
dx
(4 + x2)2
kkX∫
dx
x2√9x2 − 4
kjX∫
dx
(1− x2)3/2
k9X∫
dx
x2√x2 + 25
k8X∫
dx√x2 − 9
keX∫
dx
1 + 2x2 + x4
kdX∫
dx
(4x2 − 9)3/2
k3X∫
dx
(1− x2)2
kNX∫
dx
x2√x2 − 1
jyX∫
dx
x4√x2 + 3
jRX∫
ex√e2x + ex + 1
dx
jkX∫ √
1− 4x2 dx
jjX∫
ex√1− e2x dx
RRN
Rky
eX8 AMi2;`�iBM; _�iBQM�H 6mM+iBQMb #v S�`iB�H 6`�+iBQMb
_2+�HH i?�i � `�iBQM�H 7mM+iBQM Bb � 7mM+iBQM i?�i +�M #2 r`Bii2M �b � [mQiB2Mi Q7 irQ TQHvMQKB@
�HbX �bbmK2 i?�i f(x) =P (x)
Q(x)Bb � `�iBQM�H 7mM+iBQM- r?2`2 P (x) �M/ Q(x) �`2 TQHvMQKB�HbX A7
degP (x) < degQ(x)- i?2M f(x) Bb b�B/ iQ #2 T`QT2`X A7 degP (x) ≥ degQ(x)- i?2M f(x) Bb b�B/ iQ #2
BKT`QT2` X
q2 MQr }M/ i?2 7Q`K Q7 T�`iB�H 7`�+iBQM /2+QKTQbBiBQM Q7 � T`QT2` `�iBQM�H 7mM+iBQM f(x) =P (x)
Q(x)X
1H2K2Mi�`v �H;2#`� i2HHb mb i?�i Q(x) ?�b QMHv irQ ivT2b Q7 B``2/m+B#H2 7�+iQ`b r?B+? �`2 Q7 /2;`22
R Q` /2;`22 kX h?2`27Q`2- i?2 T�`iB�H 7`�+iBQM /2+QKTQbBiBQM Q7 f(x) +�M #2 /2i2`KBM2/ #v mbBM; i?2
7QHHQrBM; `mH2b- GBM2�` 7�+iQ` �M/ Zm�/`�iB+ 7�+iQ` `mH2bX
GBM2�` 7�+iQ` `mH2, 6Q` 2�+? 7�+iQ` Q7 i?2 7Q`K (ax+ b)m- i?2 T�`iB�H 7`�+iBQM /2+QKTQbBiBQM +QMi�BMb
i?2 7QHHQrBM; bmK Q7 m T�`iB�H 7`�+iBQMb,A1
ax+ b+
A2
(ax+ b)2+ . . .+
Am
(ax+ b)m- r?2`2 Ai (i = 1, 2, . . . ,m) �`2 +QMbi�MibX
1t�KTH2 eXRe 1p�Hm�i2∫
dx
x2 + x− 2X
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
RkR
1t�KTH2 eXRd 1p�Hm�i2∫
2x2 − 3x+ 4
(x+ 1)(x− 2)2dxX
Zm�/`�iB+ 7�+iQ` `mH2 , 6Q` 2�+? 7�+iQ` Q7 i?2 7Q`K (ax2 + bx+ c)m rBi? b2 − 4ac < 0-
i?2 T�`iB�H 7`�+iBQM /2+QKTQbBiBQM +QMi�BMb i?2 7QHHQrBM; bmK Q7 m T�`iB�H 7`�+iBQMb,A1x+B1
ax2 + bx+ c+
A2x+B2
(ax2 + bx+ c)2+ . . .+
Amx+Bm
(ax2 + bx+ c)m- r?2`2 Ai, Bi (i = 1, 2, . . . ,m) �`2 +QMbi�MibX
1t�KTH2 eXR3 1p�Hm�i2∫
3x2 + x− 2
(x− 1)(x2 + 1)dxX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rkk
1t�KTH2 eXRN 1p�Hm�i2∫
x+ 4
x2(x2 + 4)dxX
1t�KTH2 eXky 1p�Hm�i2∫
x3 − 4x
(x2 + 1)2dxX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Rkj
eX8XR AMi2;`�iBM; AKT`QT2` _�iBQM�H 6mM+iBQMb
1t�KTH2 eXkR 1p�Hm�i2∫
3x4 + 3x3 − 5x2 + x− 1
x2 + x− 2dxX
h?2 BMi2;`�M/ +�M #2 2tT`2bb2/ �b
3x4 + 3x3 − 5x2 + x− 1
x2 + x− 2= (3x2 + 1) +
1
x2 + x− 2
�M/ ?2M+2
∫3x4 + 3x3 − 5x2 + x− 1
x2 + x− 2dx =
∫(3x2 + 1) dx+
∫1
x2 + x− 2dx = x3 + x+
1
3ln |x− 1
x+ 2|+ C.
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
1t2`+Bb2 e/
q`Bi2 Qmi i?2 7Q`K Q7 i?2 T�`iB�H 7`�+iBQM /2+QKTQbBiBQMX
RX 3x− 1
(x− 3)(x+ 4)
kX 5
x(x2 − 4)
jX 2x− 3
x3 − x2
9X x2
(x+ 2)3
8X 1− x2
x3(x2 + 2)
eX 3x
(x− 1)(x2 + 6)
dX 4x3 − x
(x2 + 5)2
3X 1− 3x4
(x− 2)(x2 + 1)2
NX 1
x2
1p�Hm�i2 i?2 7QHHQrBM; BMi2;`�HbX
RyX∫
dx
x2 − 3x− 4
RRX∫
dx
x2 − 6x− 7
RkX∫
11x+ 17
2x2 + 7x− 4dx
RjX∫
5x− 5
3x2 − 8x− 3dx
R9X∫
2x2 − 9x− 9
x3 − 9xdx
R8X∫
dx
x(x2 − 1)
ReX∫
x2 − 8
x+ 3dx
RdX∫
x2 + 1
x− 1dx
R3X∫
3x2 − 10
x2 − 4x+ 4dx
RNX∫
x2
x2 − 3x+ 2dx
kyX∫
2x− 3
x2 − 3x− 10dx
kRX∫
3x+ 1
3x2 + 2x− 1dx
kkX∫
x5 + x2 + 2
x3 − xdx
kjX∫
x5 − 4x3 + 1
x3 − 4xdx
k9X∫
2x2 + 3
x(x− 1)2dx
k8X∫
3x2 − x+ 1
x3 − x2dx
keX∫
2x2 − 10x+ 4
(x+ 1)(x− 3)2dx
kdX∫
2x2 − 2x− 1
x3 − x2dx
k3X∫
x2
(x+ 1)3dx
kNX∫
2x2 + 3x+ 3
(x+ 1)3dx
jyX∫
2x2 − 1
(4x− 1)(x2 + 1)dx
jRX∫
dx
x3 + 2x
jkX∫
x3 + 3x2 + x+ 9
(x2 + 1)(x2 + 3)dx
jjX∫
x4 + 6x3 + 10x2 + x
x2 + 6x+ 10dx
Rk9
7.2}MBi2 AMi2;`�iBQM �M/ Bib �TTHB+�iBQMb
dXR �M Pp2`pB2r Q7 �`2� S`Q#H2K
:Bp2M � 7mM+iBQM f i?�i Bb +QMiBMmQmb �M/ MQMM2;�iBp2 QM �M BMi2`p�H [a, b]- }M/ i?2
�`2� #2ir22M i?2 ;`�T? Q7 f �M/ i?2 BMi2`p�H [a, b] QM i?2 x@�tBb U6B;m`2 dXRVX
6B;m`2 dXR, �`2� T`Q#H2K
dXk h?2 .2}MBiBQM Q7 �`2� �b � GBKBic aB;K� LQi�iBQM
dXkXR aB;K� LQi�iBQM
hQ bBKTHB7v Qm` +QKTmi�iBQMb- r2 rBHH #2;BM #v /Bb+mbbBM; � mb27mH MQi�iBQM 7Q` 2tT`2bbBM; H2M;i?v
bmKb BM � +QKT�+i 7Q`KX h?Bb MQi�iBQM Bb +�HH2/ bB;K� MQi�iBQM Q` bmKK�iBQM MQi�iBQM #2+�mb2 Bi
mb2b i?2 mTT2`+�b2 :`22F H2ii2`∑
iQ /2MQi2 p�`BQmb FBM/b Q7 bmKbX
Rk8
Rke
A7 f(k) Bb � 7mM+iBQM Q7 k- �M/ B7 m �M/ n �`2 BMi2;2`b bm+? i?�i m ≤ n- i?2M
n∑
k=m
f(k)
/2MQi2b i?2 bmK Q7 i?2 i2`Kb i?�i `2bmHi r?2M r2 bm#biBimi2 bm++2bbBp2 BMi2;2`b 7Q` k- bi�`iBM; rBi?
k = m �M/ 2M/BM; rBi? k = nX
1t�KTH2 dXR
∑8k=4 k
3 =
∑5k=0(−1)k(2k − 1) =
dXkXk S`QT2`iB2b Q7 amKb
h?2Q`2K dXR
U�V∑n
k=1 cak = c∑n
k=1 ak
U#V∑n
k=1(ak + bk) =∑n
k=1 ak +∑n
k=1 bk
U+V∑n
k=1(ak − bk) =∑n
k=1 ak −∑n
k=1 bk
dXkXj h?2 _2+i�M;H2 J2i?Q/ 7Q` 6BM/BM; �`2�b
PM2 �TT`Q�+? iQ i?2 �`2� T`Q#H2K Bb iQ mb2 �`+?BK2/2bǶ K2i?Q/ Q7 2t?�mbiBQM BM i?2 7QHHQrBM; r�v,
.BpB/2 i?2 BMi2`p�H [a, b] BMiQ n 2[m�H bm#BMi2`p�Hb- �M/ Qp2` 2�+? bm#BMi2`p�H +QMbi`m+i � `2+i�M;H2
i?�i 2ti2M/b 7`QK i?2 x@�tBb iQ �Mv TQBMi QM i?2 +m`p2 y = f(x) i?�i Bb �#Qp2 i?2 bm#BMi2`p�Hc i?2
T�`iB+mH�` TQBMi /Q2b MQi K�ii2` Ĝ Bi +�M #2 �#Qp2 i?2 +2Mi2`- �#Qp2 �M 2M/TQBMi- Q` �#Qp2 �Mv Qi?2`
TQBMi BM i?2 bm#BMi2`p�HX
6Q` 2�+? n- i?2 iQi�H �`2� Q7 i?2 `2+i�M;H2b +�M #2 pB2r2/ �b �M �TT`QtBK�iBQM iQ i?2 2t�+i �`2�
mM/2` i?2 +m`p2 Qp2` i?2 BMi2`p�H [a, b]X JQ`2Qp2`- Bi Bb 2pB/2Mi BMimBiBp2Hv i?�i �b n BM+`2�b2b i?2b2
�TT`QtBK�iBQMb rBHH ;2i #2ii2` �M/ #2ii2` �M/ rBHH �TT`Q�+? i?2 2t�+i �`2� �b � HBKBi U6B;m`2 dXkVX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Rkd
h?�i Bb- B7 A /2MQi2b i?2 2t�+i �`2� mM/2` i?2 +m`p2 �M/ An /2MQi2b i?2 �TT`QtBK�iBQM iQ A mbBM; n
`2+i�M;H2b- i?2M
A = limn→∞
An
q2 rBHH +�HH i?Bb i?2 `2+i�M;H2 K2i?Q/ 7Q` +QKTmiBM; AX
6B;m`2 dXk, 6BM/BM; �`2�
dXkX9 � .2}MBiBQM Q7 �`2�
/27BMBiBQM 8XR U�`2� lM/2` � *m`p2V A7 i?2 7mM+iBQM f Bb +QMiBMmQmb QM [a, b] �M/ B7
f(x) ≥ 0 7Q` �HH x BM [a, b]- i?2M i?2 �`2� A mM/2` i?2 +m`p2 y = f(x) Qp2` i?2 BMi2`p�H [a, b]
Bb /2}M2/ #v
A = limn→∞
n∑
k=1
f(x∗k)∆x.
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rk3
Ai Bb T`Q#�#Hv 2�bB2bi iQ b22 ?Qr r2 /Q i?Bb rBi? �M 2t�KTH2X aQ H2iǶb /2i2`KBM2 i?2 �`2� #2ir22M
f(x) = x2 QM [−1, 1]X AM Qi?2` rQ`/b- r2 r�Mi iQ /2i2`KBM2 i?2 �`2� Q7 i?2 b?�/2/ `2;BQM #2HQrX
6B;m`2 dXj, y = x2
aQ- H2iǶb /BpB/2 mT i?2 BMi2`p�H BMiQ 6 bm#BMi2`p�Hb �M/ mb2 i?2 7mM+iBQM p�Hm2 QM i?2 H27i Q7 2�+?
BMi2`p�H iQ /2}M2 i?2 ?2B;?i Q7 i?2 `2+i�M;H2X
6B;m`2 dX9, y = x2
6B`bi- i?2 rB/i? Q7 2�+? Q7 i?2 `2+i�M;H2b Bb . . . . . . . . . X
h?2 ?2B;?i Q7 2�+? `2+i�M;H2 Bb /2i2`KBM2/ #v i?2 7mM+iBQM p�Hm2 QM i?2 H27iX >2`2 Bb i?2 2biBK�i2/
�`2�X
A6 =
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
RkN
LQr- H2iǶb KQp2 QM iQ i?2 ;2M2`�H +�b2X q2ǶHH /BpB/2 i?2 BMi2`p�H BMiQ n bm#BMi2`p�Hb- i?2 rB/i? Q7
2�+? Q7 i?2 `2+i�M;H2b Bb . . . . . . . . . X
h?2 iQi�H �`2� An Q7 i?2 n `2+i�M;H2b rBHH #2
An = UdXRV
h�#H2 dXR #2HQr b?Qrb i?2 `2bmHi Q7 2p�Hm�iBM; UdXRV QM � +QKTmi2` 7Q` bQK2 BM+`2�bBM;Hv H�`;2
p�Hm2b Q7 nX h?2b2 +QKTmi�iBQMb bm;;2bi i?�i i?2 2t�+i �`2� Bb +HQb2 iQ . . . . . . . . . . . .X
n e Ry Ryy R-yyy Ry-yyyAn yXd yXe3 yXeee3 yXeeeee3 yXeeeeeee3
h�#H2 dXR, 2biBK�iBQM Q7 �`2�
aQ- BM+`2�bBM; i?2 MmK#2` Q7 `2+i�M;H2b BKT`Qp2b i?2 �++m`�+v Q7 i?2 2biBK�iBQM �b r2 rQmH/ ;m2bbX
G�i2` BM i?Bb +?�Ti2` r2 rBHH b?Qr i?�i
limn→∞
An =2
3.
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rjy
dXkX8 L2i aB;M2/ �`2�
A7 f Bb +QMiBMmQmb �M/ �ii�BMb #Qi? TQbBiBp2 �M/ M2;�iBp2 p�Hm2b QM [a, b]- i?2M i?2 HBKBi
limn→∞
n∑
k=1
f(x∗k)∆x
MQ HQM;2` `2T`2b2Mib i?2 �`2� #2ir22M i?2 +m`p2 y = f(x) �M/ i?2 BMi2`p�H [a, b] QM i?2 x@�tBbc `�i?2`-
Bi `2T`2b2Mib � /Bz2`2M+2 Q7 �`2�b @ě i?2 �`2� Q7 i?2 `2;BQM i?�i Bb �#Qp2 i?2 BMi2`p�H [a, b] �M/ #2HQr
i?2 +m`p2 y = f(x) KBMmb i?2 �`2� Q7 i?2 `2;BQM i?�i Bb #2HQr i?2 BMi2`p�H [a, b] �M/ �#Qp2 i?2 +m`p2
y = f(x)X q2 +�HH i?Bb i?2 M2i bB;M2/ �`2�X
6B;m`2 dX8, M2i bB;M �`2�
6Q` 2t�KTH2- BM 6B;m`2 dX8- i?2 M2i bB;M2/ �`2� #2ir22M i?2 +m`p2 y = f(x) �M/ i?2 BMi2`p�H [a, b] Bb
(AI +AIII)−AII = [ �`2� �#Qp2 [a, b]]− [ �`2� #2HQr [a, b]]
/27BMBiBQM 8Xk UL2i aB;M2/ �`2�V A7 i?2 7mM+iBQM f Bb +QMiBMmQmb QM [a, b]- i?2M i?2 M2i
bB;M2/ �`2� A #2ir22M y = f(x) �M/ i?2 BMi2`p�H [a, b] Bb /2}M2/ #v
A = limn→∞
n∑
k=1
f(x∗k)∆x.
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
RjR
dXj .2}MBi2 AMi2;`�H
dXjXR _B2K�MM amKb �M/ i?2 .2}MBi2 AMi2;`�H
AM T`2pBQmb b2+iBQM- r2 �bbmK2/ i?�i 7Q` 2�+? TQbBiBp2 MmK#2` n- i?2 BMi2`p�H [a, b] r�b bm#/BpB/2/ BMiQ
n bm#BMi2`p�Hb Q7 2[m�H H2M;i? iQ +`2�i2 #�b2b 7Q` i?2 �TT`QtBK�iBM; `2+i�M;H2bX 6Q` bQK2 7mM+iBQMb Bi
K�v #2 KQ`2 +QMp2MB2Mi iQ mb2 `2+i�M;H2b rBi? /Bz2`2Mi rB/i?bc ?Qr2p2`- B7 r2 �`2 iQ 2t?�mbi�� �M �`2�
rBi? `2+i�M;H2b Q7 /Bz2`2Mi rB/i?b- i?2M Bi Bb BKTQ`i�Mi i?�i bm++2bbBp2 bm#/BpBbBQMb �`2 +QMbi`m+i2/ BM
bm+? � r�v i?�i i?2 rB/i?b Q7 �HH i?2 `2+i�M;H2b �TT`Q�+? x2`Q �b n BM+`2�b2b U6B;m`2 dXe@H27iVX h?mb-
r2 Kmbi T`2+Hm/2 i?2 FBM/ Q7 bBim�iBQM i?�i Q++m`b BM 6B;m`2 dXe@`B;?i BM r?B+? i?2 `B;?i ?�H7 Q7 i?2
BMi2`p�H Bb M2p2` bm#/BpB/2/X A7 i?Bb FBM/ Q7 bm#/BpBbBQM r2`2 �HHQr2/- i?2 2``Q` BM i?2 �TT`QtBK�iBQM
rQmH/ MQi �TT`Q�+? x2`Q �b n BM+`2�b2/X
6B;m`2 dXe, .2}MBi2 BMi2;`�H
� T�`iBiBQM Q7 i?2 BMi2`p�H [a, b] Bb � +QHH2+iBQM Q7 TQBMib
a = x0 < x1 < x2 < · · · < xn1 < xn = b
i?�i /BpB/2b [a, b] BMiQ n bm#BMi2`p�Hb Q7 H2M;i?b
∆x1 = . . . . . . . . . ,∆x2 = . . . . . . . . . ,∆x3 = . . . . . . . . . , . . . ,∆xn = . . . . . . . . .
h?2 T�`iBiBQM Bb b�B/ iQ #2 `2;mH�` T`QpB/2/ i?2 bm#BMi2`p�Hb �HH ?�p2 i?2 b�K2 H2M;i?
∆xk = ∆x =b− a
n.
6Q` � `2;mH�` T�`iBiBQM- i?2 rB/i?b Q7 i?2 �TT`QtBK�iBM; `2+i�M;H2b �TT`Q�+? x2`Q �b n Bb K�/2 H�`;2X
aBM+2 i?Bb M22/ MQi #2 i?2 +�b2 7Q` � ;2M2`�H T�`iBiBQM- r2 M22/ bQK2 r�v iQ K2�bm`2 i?2 bBx2 Q7 i?2b2
rB/i?bX PM2 �TT`Q�+? Bb iQ H2i max∆xk /2MQi2 i?2 H�`;2bi Q7 i?2 bm#BMi2`p�H rB/i?bX h?2 K�;MBim/2
max∆xk Bb +�HH2/ i?2 K2b? bBx2 Q7 i?2 T�`iBiBQMX 6Q` 2t�KTH2- 6B;m`2 dXd b?Qrb � T�`iBiBQM Q7 i?2
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rjk
BMi2`p�H [0, 6] BMiQ 7Qm` bm#BMi2`p�Hb rBi?
6B;m`2 dXd, T�`iBiBQM Q7 (y-e)
A7 r2 �`2 iQ ;2M2`�HBx2 .2}MBiBQM dXkX9 bQ i?�i Bi �HHQrb 7Q` mM2[m�H bm#BMi2`p�H rB/i?b- r2 Kmbi
`2TH�+2 i?2 +QMbi�Mi H2M;i? ∆x #v i?2 p�`B�#H2 H2M;i? ∆xkX q?2M i?Bb Bb /QM2 i?2 bmK
n∑
k=1
f(x∗k)∆x Bb `2TH�+2/ #vn∑
k=1
f(x∗k)∆xk.
q2 �HbQ M22/ iQ `2TH�+2 i?2 2tT`2bbBQM n → ∞ #v �M 2tT`2bbBQM i?�i ;m�`�Mi22b mb i?�i i?2 H2M;i?b
Q7 �HH bm#BMi2`p�Hb �TT`Q�+? x2`QX q2 rBHH mb2 i?2 2tT`2bbBQM max∆xk → 0 7Q` i?Bb Tm`TQb2X
/27BMBiBQM � 7mM+iBQM 7 Bb b�B/ iQ #2 BMi2;`�#H2 QM � }MBi2 +HQb2/ BMi2`p�H [a, b] B7 i?2 HBKBi
limmax∆xk→0
n∑
k=1
f(x∗k)∆xk
2tBbib �M/ /Q2b MQi /2T2M/ QM i?2 +?QB+2 Q7 T�`iBiBQMb Q` QM i?2 +?QB+2 Q7 i?2 TQBMib x∗k BM
i?2 bm#BMi2`p�HbX q?2M i?Bb Bb i?2 +�b2 r2 /2MQi2 i?2 HBKBi #v i?2 bvK#QH
∫ b
af(x)dx = lim
max∆xk→0
n∑
k=1
f(x∗k)∆xk.
r?B+? Bb +�HH2/ i?2 /2}MBi2 BMi2;`�H Q7 f 7`QK a iQ bX h?2 MmK#2`b a �M/ b �`2 +�HH2/ i?2
HQr2` HBKBi Q7 BMi2;`�iBQM �M/ i?2 mTT2` HBKBi Q7 BMi2;`�iBQM - `2bT2+iBp2Hv- �M/ f(x)
Bb +�HH2/ i?2 BMi2;`�M/ X
h?2Q`2K dXk A7 � 7mM+iBQM f Bb +QMiBMmQmb QM �M BMi2`p�H [a, b]- i?2M f Bb BMi2;`�#H2
QM [a, b]- �M/ i?2 M2i bB;M2/ �`2� A #2ir22M i?2 ;`�T? Q7 f �M/ i?2 BMi2`p�H [a, b] Bb
A =
∫ b
af(x)dx.
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Rjj
1t�KTH2 dXk lb2 i?2 �`2�b b?QrM BM i?2 };m`2 iQ }M/
U�V∫ b
af(x)dx U#V
∫ c
bf(x)dx U+V
∫ c
af(x)dx U/V
∫ d
af(x)dx
aQHmiBQM
1t�KTH2 dXj aF2i+? i?2 `2;BQM r?Qb2 �`2� Bb `2T`2b2Mi2/ #v i?2 /2}MBi2 BMi2;`�H- �M/ 2p�Hm�i2 i?2
BMi2;`�H mbBM; �M �TT`QT`B�i2 7Q`KmH� 7`QK ;2QK2i`vX
U�V∫ 4
12 dx U#V
∫ 1
0
√1− x2 dx
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rj9
dXjXk S`QT2`iB2b Q7 i?2 .2}MBi2 AMi2;`�H
h?2Q`2K dXj
U�V A7 a Bb BM i?2 /QK�BM Q7 f - r2 /2}M2
∫ a
af(x)dx = 0
U#V A7 f Bb BMi2;`�#H2 QM [a, b]- i?2M r2 /2}M2
∫ b
af(x)dx = −
∫ a
bf(x)dx
1t�KTH2 dX9
U�V∫ 1
1(sin 1− x2)dx =
U#V∫ 0
1(√
1− x2)dx =
h?2Q`2K dX9 A7 f �M/ g �`2 BMi2;`�#H2 QM [a, b] �M/ B7 c Bb � +QMbi�Mi- i?2M cf -
f + g- �M/ f − g �`2 BMi2;`�#H2 QM [a, b] �M/
U�V∫ b
acf(x)dx = c
∫ b
af(x)dx
U#V∫ b
af(x) + g(x)dx =
∫ b
af(x)dx+
∫ b
ag(x)dx
U+V∫ b
af(x)− g(x)dx =
∫ b
af(x)dx−
∫ b
ag(x)dx
1t�KTH2 dX8 1p�Hm�i2∫ 1
0(5− 3
√1− x2)dx
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Rj8
h?2Q`2K dX8 A7 f Bb BMi2;`�#H2 QM � +HQb2/ BMi2`p�H +QMi�BMBM; i?2 i?`22 TQBMib a, b-
�M/ c- i?2M ∫ b
af(x)dx =
∫ c
af(x)dx+
∫ b
cf(x)dx
h?2Q`2K dXe
U�V A7 f Bb BMi2;`�#H2 QM [a, b] �M/ f(x) ≥ 0 7Q` �HH x BM [a, b]- i?2M
∫ b
af(x)dx ≥ 0
U#V A7 f �M/ g �`2 BMi2;`�#H2 QM [a, b] �M/ f(x) ≥ g(x) 7Q` �HH x BM [a, b]- i?2M
∫ b
af(x)dx ≥
∫ b
ag(x)dx
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rje
dX9 6mM/�K2Mi�H h?2Q`2K Q7 *�H+mHmb
dX9XR S�`i A Q7 i?2 6mM/�K2Mi�H h?2Q`2K Q7 *�H+mHmb
6B;m`2 dX3, �`2� mM/2` i?2 ;`�T?
�bbmK2 i?�i f Bb � MQM@M2;�iBp2 +QMiBMmQmb 7mM+iBQM QM i?2 BMi2`p�H [a, b]- i?2 �`2� A mM/2` i?2
;`�T? Q7 f Qp2` i?2 BMi2`p�H [a, b] Bb `2T`2b2Mi2/ #v i?2 /2}MBi2 BMi2;`�H
A =
∫ b
af(x)dx
U6B;m`2 dX3VX A7 A(x) /2MQi2b i?2 �`2� mM/2` i?2 ;`�T? Q7 f Qp2` i?2 BMi2`p�H [a, x]- r?2`2 x Bb �Mv
TQBMi BM i?2 BMi2`p�H [a, b] U6B;m`2 dXNV- i?2M
A′(x) = f(x) UdXkV
h?2 7QHHQrBM; 2t�KTH2 +QM}`Kb 6Q`KmH� UdXkV BM bQK2 +�b2b r?2`2 � 7Q`KmH� 7Q` A(x) +�M #2 7QmM/
mbBM; 2H2K2Mi�`v ;2QK2i`vX
6B;m`2 dXN, �`2� mM/2` i?2 ;`�T? 7`QK a iQ x
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Rjd
1t�KTH2 dXe 6Q` 2�+? Q7 i?2 7mM+iBQMb f - }M/ i?2 �`2� A(x) #2ir22M i?2 ;`�T? Q7 f �M/ i?2 BMi2`p�H
[a, x]- �M/ }M/ i?2 /2`Bp�iBp2 A′(x) Q7 i?Bb �`2� 7mM+iBQMX
aQHmiBQM
U�V f(x) = 3; a = 0
U#V f(x) = 2 + x; a = −2
h?2 T`Q+2/m`2 7Q` }M/BM; �`2�b pB� �MiB/Bz2`2MiB�iBQM Bb +�HH2/ i?2 �MiB/2`Bp�iBp2 K2i?Q/ X
_2+�T i?�i B7 A(x) Bb i?2 �`2� mM/2` i?2 ;`�T? Q7 f 7`QK a iQ x U6B;m`2 dXNV- i?2M
Ç A′(x) = f(x)
Ç A(a) = 0 Uh?2 �`2� mM/2` i?2 +m`p2 7`QK a iQ a Bb i?2 �`2� �#Qp2 i?2 bBM;H2 TQBMi a- �M/ ?2M+2
Bb x2`QXV
Ç A(b) = A Uh?2 �`2� mM/2` i?2 +m`p2 7`QK a iQ b Bb AXV
h?2 7Q`KmH� A′(x) = f(x) bi�i2b i?�i A(x) Bb �M �MiB/2`Bp�iBp2 Q7 f(x)- r?B+? BKTHB2b i?�i 2p2`v Qi?2`
�MiB/2`Bp�iBp2 Q7 f(x) QM [a, b] +�M #2 Q#i�BM2/ #v �//BM; � +QMbi�Mi iQ A(x)X �++Q`/BM;Hv- H2i
F (x) = A(x) + C
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rj3
#2 �Mv �MiB/2`Bp�iBp2 Q7 f(x)- �M/ +QMbB/2` r?�i ?�TT2Mb r?2M r2 bm#i`�+i F (a) 7`QK F (b),
F (b)− F (a) = [A(b) + C]− [A(a) + C] = A(b)−A(a) = A− 0 = A =
∫ b
af(x)dx
h?2Q`2K dXd Uh?2 6mM/�K2Mi�H h?2Q`2K Q7 *�H+mHmb- S�`i RV A7 f Bb +QMiBMmQmb
QM [a, b] �M/ F Bb �Mv �MiB/2`Bp�iBp2 Q7 f QM [a, b]- i?2M
∫ b
af(x)dx = F (b)− F (a)
1t�KTH2 dXd 1p�Hm�i2∫ 2
1xdxX
1t�KTH2 dX3
U�V 6BM/ i?2 �`2� mM/2` i?2 +m`p2 y = cosx Qp2` i?2 BMi2`p�H [0,π/2]X
U#V J�F2 � +QMD2+im`2 �#Qmi i?2 p�Hm2 Q7 i?2 BMi2;`�H
∫ π
0cosxdx
�M/ +QM}`K vQm` +QMD2+im`2 mbBM; i?2 6mM/�K2Mi�H h?2Q`2K Q7 *�H+mHmbX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
RjN
dX9Xk _2H�iBQMb?BT #2ir22M .2}MBi2 �M/ AM/2}MBi2 AMi2;`�Hb
G2i F #2 �Mv �MiB/2`Bp�iBp2 Q7 i?2 BMi2;`�M/ QM [a, b]- �M/ H2i C #2 �Mv +QMbi�Mi- i?2M
∫ b
af(x)dx = [F (b) + C]− [F (a) + C] = F (b)− F (a)
h?mb- 7Q` Tm`TQb2b Q7 2p�Hm�iBM; � /2}MBi2 BMi2;`�H r2 +�M QKBi i?2 +QMbi�Mi Q7 BMi2;`�iBQM BM
∫ b
af(x)dx = [F (x) + C]ba =
[∫f(x)dx
]b
a
r?B+? `2H�i2b i?2 /2}MBi2 �M/ BM/2}MBi2 BMi2;`�HbX
1t�KTH2 dXN
U�V∫ 9
4x2
√x dx =
U#V∫ ln 3
05ex dx =
U+V∫ 1/2−1/2
1√1− x2
dx =
dX9Xj S�`i k Q7 i?2 6mM/�K2Mi�H h?2Q`2K Q7 *�H+mHmb
h?2Q`2K dX3 A7 f Bb +QMiBMmQmb QM �M BMi2`p�H- i?2M f ?�b �M �MiB/2`Bp�iBp2 QM i?�i
BMi2`p�HX AM T�`iB+mH�`- B7 a Bb �Mv TQBMi BM i?2 BMi2`p�H- i?2M i?2 7mM+iBQM F /2}M2/
#v
F (x) =
∫ x
af(t)dt
Bb �M �MiB/2`Bp�iBp2 Q7 f c i?�i Bb- F ′(x) = f(x) 7Q` 2�+? x BM i?2 BMi2`p�H- Q` BM �M
�Hi2`M�iBp2 MQi�iBQMd
dx
[∫ x
af(t)dt
]= f(x)
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
R9y
1t�KTH2 dXRy 6BM/ d
dx
[∫ x
1t3dt
]
dX9X9 1p�Hm�iBM; .2}MBi2 AMi2;`�Hb #v am#biBimiBQM
hrQ J2i?Q/b 7Q` J�FBM; am#biBimiBQMb BM .2}MBi2 AMi2;`�Hb
� /2}MBi2 BMi2;`�H Q7 i?2 7Q`K ∫ b
a[f(u(x))u′(x)]dx,
r2 M22/ iQ �++QmMi 7Q` i?2 2z2+i i?�i i?2 bm#biBimiBQM ?�b QM i?2 x@HBKBib Q7 BMi2;`�iBQMX h?2`2 �`2
irQ r�vb Q7 /QBM; i?BbX
J2i?Q/ RX 6B`bi 2p�Hm�i2 i?2 BM/2}MBi2 BMi2;`�H
∫[f(u(x))u′(x)]dx
#v bm#biBimiBQM- �M/ i?2M mb2 i?2 `2H�iBQMb?BT
∫ b
a[f(u(x))u′(x)]dx =
[∫[f(u(x))u′(x)]dx
]b
a
,
J2i?Q/ kX J�F2 i?2 bm#biBimiBQM /B`2+iHv BM i?2 /2}MBi2 BMi2;`�H- �M/ i?2M `2TH�+2 i?2 x@HBKBib-
x = a �M/ x = b- #v +Q``2bTQM/BM; u@HBKBib- u(a) �M/ u(b)X h?Bb T`Q/m+2b � M2r /2}MBi2 BMi2;`�H
∫ b
a[f(u(x))u′(x)]dx =
∫ u(b)
u(a)[f(u)]du
i?�i Bb 2tT`2bb2/ 2MiB`2Hv BM i2`Kb Q7 uX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
R9R
1t�KTH2 dXRR lb2 i?2 irQ K2i?Q/b �#Qp2 iQ 2p�Hm�i2∫ 0
2x(x2 + 5)3dx
aQHmiBQM #v J2i?Q/ RX
aQHmiBQM #v J2i?Q/ kX
1t�KTH2 dXRk 1p�Hm�i2
U�V∫ 3/4
0
1
1− xdx U#V
∫ ln 3
0ex(1 + ex)1/2dx
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
R9k
dX9X8 AMi2;`�iBQM #v S�`ib 7Q` .2}MBi2 AMi2;`�Hb
1t�KTH2 dXRj 1p�Hm�i21∫
0
arctanx dxX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
1t2`+Bb2 d
*QKTmi2 i?2 7QHHQrBM; BMi2;`�iBQMbX
RX∫ 2
0[3x2 + x− 5] dx
kX∫ 2π
π[sinx cosx] dx
jX∫ 2
0[g(t)] dt r?2`2 g(t) =
t, 0 ≤ t < 1
sinπt, 1 ≤ t ≤ 2
9X G2i∫ 2
1f(x) dx = −4-
∫ 5
1f(x) dx = 6-
∫ 5
1g(x) dx = 8X 6BM/
U�V∫ 2
15f(x) dx
U#V∫ 5
2f(x) dx
U+V∫ 5
1[3f(x)− g(x)] dx
8X .2}M2 F (x) #v ∫ x
1[t3 + 1]dt
U�V lb2 S�`i k Q7 i?2 6mM/�K2Mi�H h?2Q`2K Q7 *�H+mHmb iQ }M/ F ′(x)X
U#V *?2+F i?2 `2bmHi BM T�`i U�V #v }`bi BMi2;`�iBM; �M/ i?2M /Bz2`2MiB�iBM;X
eX .2}M2 F (x) #v ∫ x
4
[1√t
]dt
U�V lb2 S�`i k Q7 i?2 6mM/�K2Mi�H h?2Q`2K Q7 *�H+mHmb iQ }M/ F ′(x)X
U#V *?2+F i?2 `2bmHi BM T�`i U�V #v }`bi BMi2;`�iBM; �M/ i?2M /Bz2`2MiB�iBM;X
R9j
8�TTHB+�iBQMb Q7 i?2 .2}MBi2 AMi2;`�H BM :2QK2i`v
3XR �`2� #2ir22M hrQ *m`p2b
h?2Q`2K 3XR A7 f �M/ g �`2 +QMiBMmQmb 7mM+iBQMb QM i?2 BMi2`p�H [a, b] �M/
f(x) ≥ g(x) 7Q` �HH x BM [a, b]X h?2M i?2 �`2� Q7 i?2 `2;BQM #QmM/2/ �#Qp2 #v
y = f(x)- #2HQr #v y = g(x)- QM i?2 H27i #v i?2 HBM2 x = a- �M/ QM i?2 `B;?i
#v i?2 HBM2 x = b Bb
A =
∫ b
a[f(x)− g(x)]dx. U3XRV
A = limmax∆xk→0
n∑
k=1
[f(x∗k)− g(x∗k)]∆xk =
∫ b
a[f(x)− g(x)]dx
R99
R98
1t�KTH2 3XR 6BM/ i?2 �`2� Q7 i?2 `2;BQM #QmM/2/ �#Qp2 #v y = 2x+4- #QmM/2/ #2HQr #v y = 1−x-
�M/ #QmM/2/ QM i?2 bB/2b #v i?2 HBM2 x = 0 �M/ x = 2X
1t�KTH2 3Xk 6BM/ i?2 �`2� Q7 i?2 `2;BQM 2M+HQb2/ #v y = 9− x2 �M/ y = 1 + x2X
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
R9e
1t�KTH2 3Xj 6B;m`2 b?Qrb p2HQ+Biv p2`bmb iBK2 +m`p2b 7Q` irQ `�+2 +�`b i?�i KQp2 �HQM; � bi`�B;?i
i`�+F- bi�`iBM; 7`QK `2bi �i i?2 b�K2 iBK2X :Bp2 � T?vbB+�H BMi2`T`2i�iBQM Q7 i?2 �`2� A #2ir22M i?2
+m`p2b Qp2` i?2 BMi2`p�H 0 ≤ t ≤ T.
1t�KTH2 3X9 6BM/ i?2 �`2� Q7 i?2 `2;BQM 2M+HQb2/ #v y = x- y = x2- x = 0 �M/ x = 2X
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
R9d
_2p2`bBM; i?2 _QH2b Q7 x �M/ y
h?2Q`2K 3Xk A7 w �M/ v �`2 +QMiBMmQmb 7mM+iBQMb QM i?2 BMi2`p�H (+- /) �M/ w(y) ≥
v(y) 7Q` �HH y BM (+- /)X h?2M i?2 �`2� Q7 i?2 `2;BQM #QmM/2/ QM i?2 `B;?i #v x = w(y)-
QM i?2 H27i #v x = v(y)- #2HQr #v i?2 HBM2 y = c- �M/ �#Qp2 #v i?2 HBM2 y = d Bb
A =
∫ d
c[w(y)− v(y)]dy U3XkV
A = limmax∆yk→0
n∑
k=1
[w(y∗k)− v(y∗k)]∆yk =
∫ d
c[w(y)− v(y)]dy
1t�KTH2 3X8 6BM/ i?2 �`2� Q7 i?2 `2;BQM 2M+HQb2/ #v y2 = 4x �M/ y = 2x− 4X
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
1t2`+Bb2 3�
R@9 6BM/ i?2 �`2� Q7 i?2 b?�/2/ `2;BQMbX
8@e 6BM/ i?2 �`2� Q7 i?2 b?�/2/ `2;BQM #v
U�V BMi2;`�iBM; rBi? `2bT2+i iQ x U#V BMi2;`�iBM; rBi? `2bT2+i iQ yX
R93
R9N
dX y = x2, y =√x, x =
1
4, x = 1.
3X y = x3 − 4x, y = 0, x = 0, x = 2.
NX y = cos 2x, y = 0, x = π/4, x = π/2.
RyX y = sec2 x, y = 2, x = −π/4, x = π/4.
RRX y = sin y, x = 0, y = π/4, y = 3π/4.
RkX x2 = y, x = y − 2.
RjX y = ex, y = e2x, x = 0, x = ln 2.
R9X x = 1/y, x = 0, y = 1, y = e.
R8X y = 2/(1 + x2), y = |x|.
ReX y = 1/√1− x2 , y = 2.
RdX y = x, y = 4x, y = −x+ 2.
3Xk oQHmK2b #v aHB+BM;c .BbFb �M/ q�b?2`b
h?2Q`2K 3Xj UoQHmK2 7Q`KmH�V G2i S #2 � bQHB/ #QmM/2/ #v irQ T�`�HH2H TH�M2b
T2`T2M/B+mH�` iQ i?2 x@�tBb �i x = a �M/ x = bX A7- 7Q` 2�+? x BM (�- #)- i?2 +`Qbb@
b2+iBQM�H �`2� Q7 S T2`T2M/B+mH�` iQ i?2 x@�tBb Bb A(x)- i?2M i?2 pQHmK2 Q7 i?2 bQHB/
Bb
V =
∫ b
aA(x)dx. U3XjV
V = limmax∆xk→0
n∑
k=1
A(x∗k)∆xk =
∫ b
aA(x)dx
h?2`2 Bb � bBKBH�` `2bmHi 7Q` +`Qbb b2+iBQMb T2`T2M/B+mH�` iQ i?2 y@�tBbX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
R8y
h?2Q`2K 3X9 UoQHmK2 7Q`KmH�V G2i S #2 � bQHB/ #QmM/2/ #v irQ T�`�HH2H TH�M2b
T2`T2M/B+mH�` iQ i?2 y@�tBb �i y = c �M/ y = dX A7- 7Q` 2�+? y BM (+- /)- i?2 +`Qbb@
b2+iBQM�H �`2� Q7 S T2`T2M/B+mH�` iQ i?2 y@�tBb Bb A(y)- i?2M i?2 pQHmK2 Q7 i?2 bQHB/
Bb
V =
∫ d
cA(y)dy. U3X9V
aQHB/ Q7 _2pQHmiBQM
oQHmK2 #v .BbFb T2`T2M/B+mH�` iQ i?2 X@�tBb
S`Q#H2K, G2i f #2 +QMiBMmQmb �M/ MQMM2;�iBp2 QM (�- #)- �M/ H2i R #2 i?2 `2;BQM
i?�i Bb #QmM/2/ �#Qp2 #v y = f(x)- #2HQr #v i?2 x@�tBb- �M/ QM i?2 bB/2b #v i?2
HBM2b x = a �M/ x = bX 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ Q7 `2pQHmiBQM i?�i Bb ;2M2`�i2/
#v `2pQHpBM; i?2 `2;BQM R �#Qmi i?2 X@�tBbX
q2 +�M bQHp2 i?Bb T`Q#H2K #v bHB+BM;X 6Q` i?Bb Tm`TQb2- Q#b2`p2 i?�i i?2 +`Qbb b2+iBQM Q7 i?2 bQHB/
i�F2M T2`T2M/B+mH�` iQ i?2 X@�tBb �i i?2 TQBMi x Bb � +B`+mH�` /BbF Q7 `�/Bmb f(x)X h?2 �`2� Q7 i?Bb
`2;BQM Bb
A(x) = π[f(x)]2.
h?mb- 7`QK U3XjV i?2 pQHmK2 Q7 i?2 bQHB/ Bb
V =
∫ b
aπ[f(x)]2dx. U3X8V
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
R8R
"2+�mb2 i?2 +`Qbb b2+iBQMb �`2 /BbF b?�T2/- i?2 �TTHB+�iBQM Q7 i?Bb 7Q`KmH� Bb +�HH2/ i?2 K2i?Q/
Q7 /BbFbX
1t�KTH2 3Xe 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i Bb Q#i�BM2/ r?2M i?2 `2;BQM mM/2` i?2 +m`p2 y = 3x
Qp2` i?2 BMi2`p�H (R- j) Bb `2pQHp2/ �#Qmi i?2 X@�tBbX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
R8k
oQHmK2 #v q�b?2`b S2`T2M/B+mH�` iQ i?2 X@�tBb
S`Q#H2K, G2i f �M/ g #2 +QMiBMmQmb �M/ MQMM2;�iBp2 QM [a, b]- �M/ bmTTQb2 i?�i
f(x) ≥ g(x) 7Q` �HH x BM i?2 BMi2`p�H [a, b]X G2i R #2 i?2 `2;BQM i?�i Bb #QmM/2/ �#Qp2
#v y = f(x)- #2HQr #v y = g(x)- �M/ QM i?2 bB/2b #v i?2 HBM2b x = a �M/ x = bX
6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ Q7 `2pQHmiBQM i?�i Bb ;2M2`�i2/ #v `2pQHpBM; i?2 `2;BQM
R �#Qmi i?2 X@�tBbX
q2 +�M bQHp2 i?Bb T`Q#H2K #v bHB+BM;X 6Q` i?Bb Tm`TQb2- Q#b2`p2 i?�i i?2 +`Qbb b2+iBQM Q7 i?2 bQHB/
i�F2M T2`T2M/B+mH�` iQ i?2 X@�tBb �i i?2 TQBMi x Bb i?2 �MMmH�` Q` Ǵr�b?2`@b?�T2/Ǵ- `2;BQM rBi? BMM2`
`�/Bmb g(x) �M/ Qmi2` `�/Bmb f(x)X h?2 �`2� Q7 i?Bb `2;BQM Bb
A(x) = π[f(x)]2 − π[g(x)]2 = π([f(x)]2 − [g(x)]2
)
h?mb- 7`QK U3XjV i?2 pQHmK2 Q7 i?2 bQHB/ Bb
V =
∫ b
aπ([f(x)]2 − [g(x)]2
)dx U3XeV
"2+�mb2 i?2 +`Qbb b2+iBQMb �`2 r�b?2` b?�T2/- i?2 �TTHB+�iBQM Q7 i?Bb 7Q`KmH� Bb +�HH2/ i?2 K2i?Q/
Q7 r�b?2`bX
1t�KTH2 3Xd 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i Bb Q#i�BM2/ r?2M i?2 `2;BQM #2ir22M i?2 ;`�T?b Q7
i?2 2[m�iBQMb y =√2x �M/ y = x
2 Qp2` i?2 BMi2`p�H [0, 8] Bb `2pQHp2/ �#Qmi i?2 X@�tBbX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
R8j
oQHmK2 #v .BbFb �M/ q�b?2`b T2`T2M/B+mH�` iQ i?2 Y @�tBb
h?2 K2i?Q/b Q7 /BbFb �M/ r�b?2`b ?�p2 �M�HQ;b 7Q` `2;BQMb i?�i �`2 `2pQHp2/ �#Qmi i?2 Y @�tBbX
lbBM; i?2 K2i?Q/ Q7 bHB+BM; �M/ 6Q`KmH� U3X9V- i?2 7QHHQrBM; 7Q`KmH�b 7Q` i?2 pQHmK2b Q7 i?2 bQHB/ �`2
V =
∫ d
cπ[w(y)]2dy (disks), U3XdV
V =
∫ d
cπ([w(y)]2 − [v(y)]2
)dy (washers). U3X3V
1t�KTH2 3X3 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ ;2M2`�i2/ r?2M i?2 `2;BQM 2M+HQb2/ #v x =√y- x = 0-
�M/ y = 3 Bb `2pQHp2/ �#Qmi i?2 Y @�tBbX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
R89
1t�KTH2 3XN 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ ;2M2`�i2/ r?2M i?2 `2;BQM 2M+HQb2/ #v x = 1- y =√x− 2-
y = 0- �M/ y = 1 Bb `2pQHp2/ �#Qmi i?2 Y @�tBbX
Pi?2` �t2b Q7 `2pQHmiBQM
Ai Bb TQbbB#H2 iQ mb2 i?2 K2i?Q/ Q7 /BbFb �M/ i?2 K2i?Q/ Q7 r�b?2`b iQ }M/ i?2 pQHmK2 Q7 � bQHB/
Q7 `2pQHmiBQM r?Qb2 �tBb Q7 `2pQHmiBQM Bb � HBM2 Qi?2` i?�M QM2 Q7 i?2 +QQ`/BM�i2 �t2bX AMbi2�/ Q7
/2p2HQTBM; � M2r 7Q`KmH� 7Q` 2�+? bBim�iBQM- r2 rBHH �TT2�H iQ 6Q`KmH�b U3XjV �M/ U3X9V �M/ BMi2;`�i2
�M �TT`QT`B�i2 +`Qbb@b2+iBQM�H �`2� iQ }M/ i?2 pQHmK2X
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
R88
1t�KTH2 3XRy 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i Bb Q#i�BM2/ r?2M i?2 `2;BQM #2ir22M i?2 +m`p2
y = x+ 1 �M/ y = 0 Qp2` i?2 BMi2`p�H [0, 2] Bb `Qi�i2/ �#Qmi i?2 HBM2 y = −1X
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
1t2`+Bb2 3#
RX 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i `2bmHib r?2M i?2 b?�/2/ `2;BQM Bb `2pQHp2/ �#Qmi i?2 BM/B+�i2/
�tBbX
R8e
R8d
kX 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i `2bmHib r?2M i?2 `2;BQM 2M+HQb2/ #v i?2 ;Bp2M +m`p2b Bb `2pQHp2/
�#Qmi i?2 t@�tBbX
U�V y =√25− x2, y = 3
U#V y = 9− x2, y = 0
U+V x =√y, x = y/4
U/V y = ex, y = 0, x = 0, x = ln 3
U2V y = e−2x, y = 0, x = 0, x = 1
jX 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i `2bmHib r?2M i?2 `2;BQM 2M+HQb2/ #v i?2 ;Bp2M +m`p2b Bb `2pQHp2/
�#Qmi i?2 v@�tBbX
U�V y = csc y, y = π/4, y = 3π/4, x = 0
U#V y = x2, x = y2
U+V x = y2, x = y + 2
U/V x = 1− y2, x = 2 + y2, y = −1, y = 1
U2V y = lnx, x = 0, y = 0, y = 1
9X 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i `2bmHib r?2M i?2 `2;BQM 2M+HQb2/ #v y =√x, y = 0- �M/ x = 9
Bb `2pQHp2/ �#Qmi i?2 HBM2 x = 9X
8X 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i `2bmHib r?2M i?2 `2;BQM BM S`Q#H2K 9 Bb `2pQHp2/ �#Qmi i?2
HBM2 x = 9X
eX 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i `2bmHib r?2M i?2 `2;BQM 2M+HQb2/ #v x = y2 �M/ x = y Bb
`2pQHp2/ �#Qmi i?2 HBM2 y = −1X
dX 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i `2bmHib r?2M i?2 `2;BQM BM S`Q#H2K e Bb `2pQHp2/ �#Qmi i?2
HBM2 x = −1X
3X 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i `2bmHib r?2M i?2 `2;BQM 2M+HQb2/ #v y = x2 �M/ y = x3 Bb
`2pQHp2/ �#Qmi i?2 HBM2 x = 1X
NX 6BM/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i `2bmHib r?2M i?2 `2;BQM BM i?2 T`Q#H2K Q7 Bi2K 3 Bb `2pQHp2/
�#Qmi i?2 HBM2 y = −1X
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
R83
3Xj oQHmK2b #v *vHBM/`B+�H a?2HHb
h?2Q`2K 3X8 UoQHmK2 #v +vHBM/`B+�H b?2HHb �#Qmi i?2 u@�tBbV G2i f #2
+QMiBMmQmb �M/ MQMM2;�iBp2 QM (�- #) �M/ H2i R #2 i?2 `2;BQM i?�i Bb #QmM/2/ �#Qp2
#v y = f(x)- #2HQr #v i?2 X@�tBb- �M/ QM i?2 bB/2b #v i?2 HBM2b x = a �M/ x = bX
h?2M i?2 pQHmK2 V Q7 i?2 bQHB/ Q7 `2pQHmiBQM i?�i Bb ;2M2`�i2/ #v `2pQHpBM; i?2
`2;BQM R �#Qmi i?2 Y @�tBb Bb ;Bp2M #v
V =
∫ b
a2πxf(x)dx. U3XNV
V = limmax∆xk→0
n∑
k=1
2πx∗kf(x∗k)∆xk =
∫ b
a2πxf(x)dx.
1t�KTH2 3XRR lb2 +vHBM/`B+�H b?2HHb iQ }M/ i?2 pQHmK2 Q7 i?2 bQHB/ ;2M2`�i2/ r?2M i?2 `2;BQM 2M+HQb2/
#2ir22M y = x2- x = 1- x = 2 �M/ i?2 X@�tBb Bb `2pQHp2/ �#Qmi i?2 Y @�tBbX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
R8N
1t�KTH2 3XRk lb2 +vHBM/`B+�H b?2HHb iQ }M/ i?2 pQHmK2 Q7 i?2 bQHB/ ;2M2`�i2/ r?2M i?2 `2;BQM _ BM
i?2 }`bi [m�/`�Mi 2M+HQb2/ #2ir22M y = x �M/ y = x2 Bb `2pQHp2/ �#Qmi i?2 Y @�tBbX
1t�KTH2 3XRj lb2 +vHBM/`B+�H b?2HHb iQ }M/ i?2 pQHmK2 Q7 i?2 bQHB/ ;2M2`�i2/ r?2M i?2 `2;BQM R
mM/2` y =√x Qp2` i?2 BMi2`p�H [0, 1] Bb `2pQHp2/ �#Qmi
RX HBM2 y = −1X kX t@�tBbX jX v@�tBbX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
1t2`+Bb2 3+
RX lb2 +vHBM/`B+�H b?2HHb iQ }M/ i?2 pQHmK2 Q7 i?2 bQHB/ ;2M2`�i2/ r?2M i?2 b?�/2/ `2;BQM Bb `2pQHp2/
�#Qmi i?2 BM/B+�i2/ �tBbX
kX lb2 +vHBM/`B+�H b?2HHb iQ }M/ i?2 pQHmK2 Q7 i?2 bQHB/ ;2M2`�i2/ r?2M i?2 `2;BQM 2M+HQb2/ #v i?2
;Bp2M +m`p2b Bb `2pQHp2/ �#Qmi i?2 v@�tBbX
U�V y = x3, x = 1, y = 0
U#V y =√x, x = 4, x = 9, y = 0
U+V y = 1/x, y = 0, x = 1, x = 3
U/V y = cos(x2), x = 0, x = 12
√π, y = 0
U2V y = 2x− 1, y = −2x+ 3, x = 2
U7V y = 2x− x2, y = 0
Rey
ReR
jX lb2 +vHBM/`B+�H b?2HHb iQ }M/ i?2 pQHmK2 Q7 i?2 bQHB/ ;2M2`�i2/ r?2M i?2 `2;BQM 2M+HQb2/ #v i?2
;Bp2M +m`p2b Bb `2pQHp2/ �#Qmi i?2 t@�tBbX
U�V y2 = x, y = 1, x = 0
U#V x = 2y, y = 2, y = 3, x = 0
U+V y = x2, x = 1, y = 0
U/V xy = 4, x+ y = 5
9X lbBM; i?2 K2i?Q/ Q7 +vHBM/`B+�H b?2HHb- b2i mT #mi /Q MQi 2p�Hm�i2 �M BMi2;`�H 7Q` i?2 pQHmK2
Q7 i?2 bQHB/ ;2M2`�i2/ r?2M i?2 `2;BQM R Bb `2pQHp2/ �#Qmi U�V i?2 HBM2 x = 1 �M/ U"V i?2 HBM2
y = −1X
U�V R Bb i?2 `2;BQM #QmM/2/ #v i?2 ;`�T?b Q7 y = x- y = 0- �M/ x = 1X
U#V R Bb i?2 `2;BQM BM i?2 }`bi [m�/`�Mi #QmM/2/ #v i?2 ;`�T?b Q7 y =√1− x2, y = 0 �M/
x = 0X
8X lb2 +vHBM/`B+�H b?2HHb iQ }M/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i Bb ;2M2`�i2/ r?2M i?2 `2;BQM i?�i Bb
2M+HQb2/ #v y = 1/x3, x = 1, x = 2, y = 0 Bb 2pQHp2/ �#Qmi i?2 HBM2 x = −1X
eX lb2 +vHBM/`B+�H b?2HHb iQ }M/ i?2 pQHmK2 Q7 i?2 bQHB/ i?�i Bb ;2M2`�i2/ r?2M i?2 `2;BQM i?�i Bb
2M+HQb2/ #v y = x3, y = 1, x = 0 Bb 2pQHp2/ �#Qmi i?2 HBM2 y = 1X
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rek
3X9 AKT`QT2` AMi2;`�Hb
Ai Bb �bbmK2/ BM i?2 /2}MBiBQM Q7 i?2 /2}MBi2 BMi2;`�H∫ ba f(x) dx i?�i [a, b] Bb � }MBi2 BMi2`p�H �M/ i?�i
i?2 HBKBi i?�i /2}M2b i?2 BMi2;`�H 2tBbibc i?�i Bb- i?2 7mM+iBQM f Bb BMi2;`�#H2X
Pm` K�BM Q#D2+iBp2 Bb iQ 2ti2M/ i?2 +QM+2Ti Q7 /2}MBi2 BMi2;`�Hb 7Q` BM}MBi2 BMi2`p�Hb Q7 BMi2;`�iBQM
�M/ 7Q` BMi2;`�M/b rBi? p2`iB+�H �bvKTiQi2b rBi?BM i?2 BMi2`p�H Q7 BMi2;`�iBQMX A7 � 7mM+iBQM f ?�b �
p2`iB+�H �bvKTiQi2- i?2M f Bb b�B/ iQ ?�p2 �M BM}MBi2 /Bb+QMiBMmBiv X
�M BMi2;`�H Qp2` �M BM}MBi2 BMi2`p�H Q7 BMi2;`�iBQM Q` �M BMi2;`�H rBi? �M BM}MBi2 /Bb+QMiBMmBiv rBHH
#2 +�HH2/ �M BKT`QT2` BMi2;`�H X
h?2`2 �`2 i?`22 ivT2b Q7 BKT`QT2` BMi2;`�Hb,
RX AKT`QT2` BMi2;`�Hb rBi? BM}MBi2 BMi2`p�Hb Q7 BMi2;`�iBQMX
kX AKT`QT2` BMi2;`�Hb rBi? BM}MBi2 /Bb+QMiBMmBiB2b BM i?2 BMi2`p�H Q7 BMi2;`�iBQMX
jX AKT`QT2` BMi2;`�Hb rBi? BM}MBi2 /Bb+QMiBMmBiB2b Qp2` BM}MBi2 BMi2`p�Hb Q7 BMi2;`�iBQMX
1t�KTH2 3XR9 .2i2`KBM2 B7 2�+? Q7 i?2 7QHHQrBM; BMi2;`�Hb Bb BKT`QT2`X A7 bQ- bT2+B7v Bib ivT2X
RX∫ +∞
0
1
1− x2dx kX
∫ +∞
−∞x3 dx
jX∫ π
0sec2 θ dθ 9X
∫ 3
−3
x
x2 + x+ 1dx
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Rej
3X9XR AMi2;`�Hb Qp2` AM}MBi2 AMi2`p�Hb ,
amTTQb2 r2 �`2 BMi2`2bi2/ BM i?2 �`2� A Q7 i?2 `2;BQM i?�i HB2b #2HQr i?2 +m`p2 y = 1/x2 �M/ �#Qp2
i?2 BMi2`p�H [1,+∞) QM i?2 t@�tBbX G2i mb #2;BM #v +�H+mH�iBM; i?2 TQ`iBQM Q7 i?2 �`2� i?�i HB2b �#Qp2
� }MBi2 BMi2`p�H [1, b]- r?2`2 b > 1 Bb �`#Bi`�`vX h?�i �`2� Bb∫ b1
dxx = 1− 1
b
A7 r2 MQr �HHQr b iQ BM+`2�b2 bQ i?�i b → +∞- i?2M i?2 TQ`iBQM Q7 i?2 �`2� Qp2` i?2 BMi2`p�H [1, b] rBHH
#2;BM iQ }HH Qmi i?2 �`2� Qp2` i?2 2MiB`2 BMi2`p�H [1,+∞)- �M/ ?2M+2 r2 +�M `2�bQM�#Hv /2}M2 i?2 �`2�
A mM/2` y = 1/x2 Qp2` i?2 BMi2`p�H [1,+∞) iQ #2 A =∫∞1
dxx = limb→+∞
∫ b1
dxx = limb→+∞(1− 1
b ) = 1
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Re9
.27BMBiBQM h?2 BKT`QT2` BMi2;`�H Q7 7 Qp2` i?2 BMi2`p�H [a,+∞) Bb /2}M2/ iQ #2
∫ ∞
af(x)dx = lim
b→∞
∫ b
af(x)dx,
h?2 BMi2;`�H Bb b�B/ iQ +QMp2`;2 B7 i?2 HBKBi 2tBbib �M/ /Bp2`;2 B7 Bi /Q2b MQiX
h?2 BKT`QT2` BMi2;`�H Q7 7 Qp2` i?2 BMi2`p�H (−∞, b] Bb /2}M2/ iQ #2
∫ b
−∞f(x)dx = lim
a→−∞
∫ b
af(x)dx,
h?2 BMi2;`�H Bb b�B/ iQ +QMp2`;2 B7 i?2 HBKBi 2tBbib �M/ /Bp2`;2 B7 Bi /Q2b MQiX
h?2 BKT`QT2` BMi2;`�H Q7 7 Qp2` i?2 BMi2`p�H (−∞,+∞) Bb /2}M2/ �b
∫ ∞
−∞f(x)dx =
∫ c
−∞f(x)dx+
∫ ∞
cf(x)dx
r?2`2 c Bb �Mv `2�H MmK#2`X h?2 BKT`QT2` BMi2;`�H Bb b�B/ iQ +QMp2`;2 B7 #Qi? i2`Kb +QMp2`;2
�M/ /Bp2`;2 B7 2Bi?2` i2`K /Bp2`;2bX
1t�KTH2 3XR8 *QKTmi2∫ +∞
1
1
x3dxX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Re8
1t�KTH2 3XRe *QKTmi2∫ +∞
0cosx dxX
1t�KTH2 3XRd *QKTmi2∫ ∞
1
lnx
xdxX
1t�KTH2 3XR3 *QKTmi2∫ 1
−∞
1
3− 2xdxX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Ree
1t�KTH2 3XRN *QKTmi2∫ ∞
−∞
x
x2 + 1dxX
3X9Xk AMi2;`�Hb r?Qb2 AMi2;`�M/b ?�p2 AM}MBi2 .Bb+QMiBMmBiB2b,
G2i mb +QMbB/2` i?2 +�b2 r?2`2 f Bb MQMM2;�iBp2 QM [a, b]- bQ r2 +�M BMi2`T`2i i?2 BKT`QT2` BMi2;`�H∫ ba f(x) dx �b i?2 �`2� Q7 i?2 `2;BQMX h?2 T`Q#H2K Q7 }M/BM; i?2 �`2� Q7 i?Bb `2;BQM Bb +QKTHB+�i2/ #v
i?2 7�+i i?�i Bi 2ti2M/b BM/2}MBi2Hv BM i?2 TQbBiBp2 v@/B`2+iBQMX >Qr2p2`- BMbi2�/ Q7 i`vBM; iQ }M/ i?2
2MiB`2 �`2� �i QM+2- r2 +�M T`Q+22/ BM/B`2+iHv #v +�H+mH�iBM; i?2 TQ`iBQM Q7 i?2 �`2� Qp2` i?2 BMi2`p�H
[a, k]- r?2`2 a ≤ k < b- �M/ i?2M H2iiBM; k �TT`Q�+? b iQ }HH Qmi i?2 �`2� Q7 i?2 2MiB`2 `2;BQMX
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Red
.27BMBiBQM A7 f Bb +QMiBMmQmb QM i?2 BMi2`p�H [a, b]- 2t+2Ti 7Q` �M BM}MBi2 /Bb+QMiBMmBiv �i
b- i?2M i?2 BKT`QT2` BMi2;`�H Q7 7 Qp2` i?2 BMi2`p�H [a, b] Bb /2}M2/ �b
∫ b
af(x)dx = lim
k→b−
∫ k
af(x)dx,
h?2 BMi2;`�H Bb b�B/ iQ +QMp2`;2 B7 i?2 BM/B+�i2/ HBKBi 2tBbib �M/ /Bp2`;2 B7 Bi /Q2b MQiX
A7 f Bb +QMiBMmQmb QM i?2 BMi2`p�H [a, b]- 2t+2Ti 7Q` �M BM}MBi2 /Bb+QMiBMmBiv �i a- i?2M i?2
BKT`QT2` BMi2;`�H Q7 7 Qp2` i?2 BMi2`p�H [a, b] Bb /2}M2/ �b
∫ b
af(x)dx = lim
k→a+
∫ b
kf(x)dx,
h?2 BMi2;`�H Bb b�B/ iQ +QMp2`;2 B7 i?2 BM/B+�i2/ HBKBi 2tBbib �M/ /Bp2`;2 B7 Bi /Q2b MQiX
A7 7 Bb +QMiBMmQmb QM i?2 BMi2`p�H [a, b]- 2t+2Ti 7Q` �M BM}MBi2 /Bb+QMiBMmBiv �i � TQBMi c BM
(a, b)- i?2M i?2 BKT`QT2` BMi2;`�H Q7 f Qp2` i?2 BMi2`p�H [a, b] Bb /2}M2/ �b
∫ b
af(x)dx =
∫ c
af(x)dx+
∫ b
cf(x)dx,
r?2`2 i?2 irQ BMi2;`�Hb QM i?2 `B;?i bB/2 �`2 i?2Kb2Hp2b BKT`QT2`X h?2 BKT`QT2` BMi2;`�H
QM i?2 H27i bB/2 Bb b�B/ iQ +QMp2`;2 B7 #Qi? i2`Kb QM i?2 `B;?i bB/2 +QMp2`;2 �M/ /Bp2`;2 B7
2Bi?2` i2`K QM i?2 `B;?i bB/2 /Bp2`;2bX
1t�KTH2 3Xky *QKTmi21∫
0
1
1− xdxX
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Re3
1t�KTH2 3XkR *QKTmi23∫
0
dx
(x− 1)2/3X
h?2 7QHHQrBM; 2t�KTH2 Bb �M BKT`QT2` BMi2;`�H rBi? BM}MBi2 /Bb+QMiBMmBiv Qp2` BM}MBi2 BMi2`p�Hb Q7
BMi2;`�iBQMX _2K�`F i?�i Bi +QK#BM2b i?2 }`bi �M/ b2+QM/ ivT2 Q7 BKT`QT2` BMi2;`�HX qBi? i?Bb ivT2-
r2 b?QmH/ b2T�`�i2 i?2 BKBT`QT2` BMi2;`�H BMiQ i?2 }`bi �M/ b2+QM/ ivT2bX h?2M- i?2 +QMp2`;2M+2 rBHH
#2 +QMbB2`2/ mbBM; i?2 K2MiBQM2/ K2i?Q/bX
1t�KTH2 3Xkk *QKTmi2∞∫
−∞
dx
(x− 1)2/3X
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
1t2`+Bb2 3/
1p�Hm�i2 i?2 7QHHQrBM; BMi2;`�Hb B7 i?2v +QMp2`;2X
RX∫ 4
3
1
(x− 3)2dx
kX+∞∫
1
dx
x1.001
jX0∫
−∞
θeθdθ
9X+∞∫
2
2
v2 − vdv
8X+∞∫
0
sinπx dx
eX+∞∫
−∞
1
1 + 4x2dx
dX+∞∫
0
xdx√x+ 1
3X0∫
−∞
dx
x2 + 4
NX+∞∫
1
dx
x4 + x2
RyX+∞∫
−∞
3xdx
RRX4∫
−1
dx√|x|
RkX∫ 2
0
x
x2 − 1dx
RjX+∞∫
0
e2xdx
R9X1∫
0
dx√1− x2
R8X2∫
0
dx
(x− 1)1/3
ReX2∫
1
dx
(2− x)3/4
RdXπ/2∫
0
xdx
sinx2
R3X−2∫
−∞
2
x2 − 1dx
RNX1∫
0
lnx
xdx
kyX+∞∫
0
dx
(1 + x)√x
kRX1∫
0
θ + 1√θ2 + 2θ
dθ
kkX∫ ∞
2
1
x2 + 4dx
kjX∫ 3
1x(x2 − 4)−3dx
k9X∫ 2
0
2x+ 1
x2 + x− 6dx
k8X∫ 4
0
ln√x√x
dx
keX∫ 4
2
x3√x− 2
dx
kdX∫ 3
−1
13√xdx
k3X∫ 1
−1
1√|x|
dx
kNX∫ 3
0
1
x2 + 2x− 3dx
jyX∫ 2
−1
1
x2cos
1
xdx
jRX∫ 2
−1
1
x2 − x− 2dx
jkX∫ 1
0
1√1− x2
dx
jjX∫ ∞
0xe−xdx
j9X∫ π/2
0sec2 xdx
j8X∫ 1
0x lnxdx
jeX∫ 4
0
1
(4− x)3/2dx
jdX∫ ∞
−∞
x
(x2 + 3)2dx
ReN
Rdy
j3X∫ ∞
−∞
|1 + x|x2 + 1
dx
jNX∫ 0
−∞
1
(x− 8)2/3dx
9yX∫ 0
−∞
1
(1− x)5/2dx
9RX∫ 0
−∞e3xdx
9kX∫ ∞
1
1√x(1 + e
√x)
2dx
9jX∫ ∞
0e−x cosxdx
99X∫ ∞
−1
x
1 + x2dx
98X∫ 4
2(x− 3)−7dx
9eX∫ ∞
0cosxdx
9dX∫ ∞
−∞
1
ex + e−xdx
93X∫ 0
−∞
1
2x2 + 2x+ 1dx
9NX∫ −1
−∞
x√1 + x2
dx
8yX∫ ∞
0
1
e2x + exdx
8RX∫ 0
−∞
ex
3− 2exdx
�TTHB+�iBQMb �M/ *QM+2Tib,
8kX 6BM/ i?2 �`2� Q7 i?2 `2;BQM #2ir22M i?2 x@�tBb �M/ i?2 +m`p2 8/(x2 − 4), x > 4.
8jX G2i R #2 i?2 `2;BQM iQ i?2 `B;?i Q7 x = 1 i?�i Bb #QmM/2/ #v i?2 x@�tBb �M/ i?2 +m`p2 y = 1/xX
q?2M i?Bb `2;BQM Bb `2pQHp2/ �#Qmi i?2 x@�tBb- Bi ;2M2`�i2b � bQHB/ r?Qb2 bm`7�+2 Bb FMQrM �b
:�#`B2HǶb >Q`M U7Q` `2�bQMb i?�i b?QmH/ #2 +H2�` 7`QK i?2 �++QKT�MvBM; };m`2 3XR VX a?Qr i?�i
i?2 bQHB/ ?�b � }MBi2 pQHmK2 #mi Bib bm`7�+2 ?�b �M BM}MBi2 �`2�X LQi2, Ai ?�b #22M bm;;2bi2/
i?�i B7 QM2 +QmH/ b�im`�i2 i?2 BMi2`BQ` Q7 i?2 bQHB/ rBi? T�BMi �M/ �HHQr Bi iQ b22T i?`Qm;? iQ
i?2 bm`7�+2- i?2M QM2 +QmH/ T�BMi �M BM}MBi2 bm`7�+2 rBi? � }MBi2 �KQmMi Q7 T�BMiX q?�i /Q vQm
i?BMF\
h`m2 Q` 6�Hb2, .2i2`KBM2 r?2i?2` i?2 7QHHQrBM; bi�i2K2Mib �`2 i`m2 Q` 7�Hb2X 1tTH�BM vQm`
�Mbr2`X
89X∫ ∞
1x−4/3 dx +QMp2`;2b iQ jX
88X A7 f Bb +QMiBMmQmb QM [a,+∞) �M/ limx→+∞
f(x) = 1- i?2M∫ +∞
af(x)dx +QMp2`;2bX
8eX∫ 2
1
1
x(x− 3)dx Bb �M BKT`QT2` BMi2;`�HX
8dX∫ 1
−1
1
x3dx = 0X
6B;m`2 3XR, :�#`B2HǶb >Q`M
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
9.Bz2`2MiB�H 1[m�iBQMb
AM i?Bb +?�Ti2`- r2 BMi`Q/m+2 irQ K2i?Q/b 7Q` bQHpBM; bQK2 7Q`K Q7 i?2 }`bi Q`/2` Q7 /Bz2`2MiB�H
2[m�iBQMb UP.1bVX 6B`bi- r2 BMi`Q/m+2 bQK2 #�bB+ /2}MBiBQMb Q7 P.1bX q2- i?2M- bQHp2 i?2 T�`iB+mH�`
P.1b BM i?2 7Q`Kb Q7 a2T�`�#H2 2[m�iBQMb �M/ GBM2�` }`bi Q`/2` P.1b X G�biHv- bQK2 2t�KTH2b
Q7 HBM2�` }`bi Q`/2` P.1bX
NXR AMi`Q/m+iBQM iQ P`/BM�`v .Bz2`2MiB�H 1[m�iBQMb
*QMbB/2` i?2 2[m�iBQM- y = 2x3 − 2x2 + 5X "v /Bz2`2MiB�iBQM- Bi +�M #2 b?QrM i?�i
dy
dx= 6x2 − 4x. UNXRV
aBKBH�`Hv- 7Q` � 7mM+iBQM p(x) = 10000e−0.04x- r2 ?�p2
p′(x) = −400e−0.04x. UNXkV
h?2b2 2[m�iBQMb �`2 2t�KTH2 Q7 /Bz2`2MiB�H 2[m�iBQMb X
AM ;2M2`�H- �M 2[m�iBQM Bb � /Bz2`2MiB�H 2[m�iBQM B7 Bi BMpQHp2b �M mMFMQrM 7mM+iBQM �M/ QM2 Q`
KQ`2 Q7 Bib /2`Bp�iBp2bX Pi?2` 2t�KTH2b Q7 /Bz2`2MiB�H 2[m�iBQMb �`2
dy
dx= ky, y′′ − xy′ + x2 = 5,
dy
dx= 2xy
h?2 }`bi �M/ i?B`/ 2[m�iBQMb �`2 +�HH2/ }`bi@Q`/2` 2[m�iBQMb #2+�mb2 2�+? BMpQHp2b � }`bi /2`Bp�@
iBp2 #mi MQ ?B;?2` /2`Bp�iBp2X h?2 b2+QM/ 2[m�iBQM Bb +�HH2/ � b2+QM/@Q`/2` 2[m�iBQM #2+�mb2 Bi
BMpQHp2b � b2+QM/ /2`Bp�iBp2 �M/ MQ ?B;?2` /2`Bp�iBp2bX AM ;2M2`�H- i?2 Q`/2` Q7 � /Bz2`2MiB�H 2[m�iBQM
Bb i?2 Q`/2` Q7 i?2 ?B;?2bi /2`Bp�iBp2 i?�i Bi +QMi�BMbX
RdR
Rdk
NXk :2M2`�H �M/ S�`iB+mH�` aQHmiBQMb
� bQHmiBQM Q7 /Bz2`2MiB�H 2[m�iBQM Bb i?2 7mM+iBQM r?B+? K�i+?2b i?2 /Bz2`2MiB�H 2[m�iBQMX
1t�KTH2 NXR a?Qr i?�i i?2 7mM+iBQM y = ex Bb � bQHmiBQM Q7
dy
dx− y = 0
1t�KTH2 NXk a?Qr i?�i- 7Q` �Mv +QMbi�Mi C- i?2 7mM+iBQM y = ex − x+ C Bb � bQHmiBQM Q7
dy
dx= ex − 1
_2K�`F,
Ç h?2 ;2M2`�H bQHmiBQM Q7 � /Bz2`2MiB�H 2[m�iBQM Bb � bQHmiBQM i?�i +QMi�BMb �HH TQbbB#H2 bQHmiBQMbX
h?2 ;2M2`�H bQHmiBQM �Hr�vb +QMi�BMb �M �`#Bi`�`v +QMbi�MiX
Ç h?2 T�`iB+mH�` bQHmiBQM Q7 � /Bz2`2MiB�H 2[m�iBQM Bb � bQHmiBQM i?�i b�iBb}2b i?2 BMBiB�H +QM/BiBQM
Q7 i?2 2[m�iBQMX � }`bi@Q`/2` BMBiB�H p�Hm2 T`Q#H2K Bb � }`bi@Q`/2` /Bz2`2MiB�H 2[m�iBQM
y′ = f(x, y) r?Qb2 bQHmiBQM Kmbi b�iBb7v �M BMBiB�H +QM/BiBQM y(x0) = y0X
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Rdj
1t�KTH2 NXj 6BM/ i?2 T�`iB+mH�` bQHmiBQM Q7
dy
dx= ex − 1, y(0) = 1.
1t�KTH2 NX9 a?Qr i?�i i?2 7mM+iBQM
y = (x+ 1)− 1
3ex
Bb � bQHmiBQM iQ i?2 }`bi Q`/2` BMBiB�H@p�Hm2 T`Q#H2K
dy
dx= y − x, y(0) = 2/3.
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rd9
NXj a2T�`�#H2 1[m�iBQMb
q2 rBHH MQr +QMbB/2` � K2i?Q/ Q7 bQHmiBQM i?�i +�M Q7i2M #2 �TTHB2/ iQ }`bi@Q`/2` 2[m�iBQMb i?�i �`2
2tT`2bbB#H2 BM i?2 7Q`K
h(y)dy
dx= g(x). UNXjV
am+? }`bi@Q`/2` 2[m�iBQMb �`2 b�B/ iQ #2 b2T�`�#H2 X h?2 M�K2 �b2T�`�#H2� �`Bb2b 7`QK i?2 7�+i i?�i
UNXjV +�M #2 `2r`Bii2M BM i?2 /Bz2`2MiB�H 7Q`K
h(y)dy = g(x)dx UNX9V
BM r?B+? i?2 2tT`2bbBQMb BMpQHpBM; x �M/ y �TT2�` QM QTTQbBi2 bB/2bX hQ KQiBp�i2 � K2i?Q/ 7Q` bQHpBM;
b2T�`�#H2 2[m�iBQMb- �bbmK2 i?�i h(y) �M/ g(x) �`2 +QMiBMmQmb 7mM+iBQMb Q7 i?2B` `2bT2+iBp2 p�`B�#H2b-
�M/ H2i H(y) �M/ G(x) /2MQi2 �MiB/2`Bp�iBp2b Q7 h(y) �M/ g(x)- `2bT2+iBp2HvX *QMbB/2` i?2 `2bmHib B7
r2 BMi2;`�i2 #Qi? bB/2b Q7 UNX9V- i?2 H27i bB/2 rBi? `2bT2+i iQ y �M/ i?2 `B;?i bB/2 rBi? `2bT2+i iQ xX
q2 i?2M ?�p2
∫h(y)dy =
∫g(x)dx, UNX8V
Q`- 2[mBp�H2MiHv-
H(y) = G(x) + C UNXeV
r?2`2 C /2MQi2b � +QMbi�MiX q2 +H�BK i?�i � /Bz2`2MiB�#H2 7mM+iBQM y = y(x) Bb � bQHmiBQM iQ UNXjV B7
�M/ QMHv B7 y b�iBb}2b UNXeV 7Q` bQK2 +?QB+2 Q7 i?2 +QMbi�Mi *X
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Rd8
1t�KTH2 NX8 q`Bi2 i?2b2 }`bi@Q`/2` /Bz2`2MiB�H 2[m�iBQM BM i?2 b2T�`�#H2 7Q`KX
1[m�iBQM 6Q`K h(y) g(x)
dy
dx=
x
y
dy
dx= x2y3
dy
dx= y
dy
dx= y − y
x
1t�KTH2 NXe 6BM/ i?2 ;2M2`�H bQHmiBQM Q7
dy
dx=
x
y.
1t�KTH2 NXd 6BM/ i?2 ;2M2`�H bQHmiBQM Q7
dy
dx= yex.
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rde
1t�KTH2 NX3 6BM/ i?2 ;2M2`�H bQHmiBQM Q7
dy
dx=
√xy .
1t�KTH2 NXN 6BM/ i?2 ;2M2`�H bQHmiBQM Q7
dy
dx=
xy + y
xy − x
1t�KTH2 NXRy aQHp2 i?2 BMBiB�H p�Hm2 T`Q#H2K
dy
dx= −4xy2, y(0) = 1.
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
Rdd
1t�KTH2 NXRR aQHp2 i?2 BMBiB�H p�Hm2 T`Q#H2K
yy′ − (x2 + 1) = 0, y(4) = 2.
1t�KTH2 NXRk aQHp2 i?2 BMBiB�H p�Hm2 T`Q#H2K
(4y − cos y)dy
dx− 3x2 = 0, y(0) = 0.
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
Rd3
NX9 GBM2�` 1[m�iBQMb
� }`bi@Q`/2` /Bz2`2MiB�H 2[m�iBQM Bb +�HH2/ HBM2�` B7 Bi Bb 2tT`2bbB#H2 BM i?2 7Q`K
dy
dx+ p(x) · y = q(x). UNXdV
aQK2 2t�KTH2b Q7 }`bi@Q`/2` HBM2�` /Bz2`2MiB�H 2[m�iBQMb �`2
dy
dx= x3 − xy,
dy
dx+ x2y = ex,
dy
dx+ (sinx)y + x3 = 0,
dy
dx+ 5y + 2 = 0.
q2 rBHH �bbmK2 i?�i i?2 7mM+iBQMb p(x) �M/ q(x) BM UNXdV �`2 +QMiBMmQmb �M/ r2 rBHH HQQF 7Q` � ;2M2`�H
bQHmiBQM i?�i Bb p�HB/ QM i?�i BMi2`p�HX PM2 K2i?Q/ 7Q` /QBM; i?Bb Bb #�b2/ QM i?2 Q#b2`p�iBQM i?�i B7
r2 /2}M2 i?2 7mM+iBQM I = I(x) #v
I = e∫p(x)dx. UNX3V
i?2M
dI
dx= e
∫p(x)dx · d
dx
∫p(x)dx = I · p(x).
h?mb-
d
dx(Iy) = I
dy
dx+
dI
dxy = I
dy
dx+ Ip(x)y. UNXNV
A7 UNXdV Bb KmHiBTHB2/ i?`Qm;? #v I- Bi #2+QK2b
Idy
dx+ Ip(x) · y = Iq(x).
*QK#BM2 i?Bb rBi? UNXNV- r2 ?�p2
d
dx(Iy) = Iq(x).
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
RdN
h?Bb 2[m�iBQM +�M #2 bQHp2/ 7Q` y #v BMi2;`�iBM; #Qi? bB/2b rBi? `2bT2+i iQ x �M/ i?2M /BpB/BM; i?`Qm;?
#v I iQ Q#i�BM
y =1
I(x)
∫I(x)q(x)dx
r?B+? Bb � ;2M2`�H bQHmiBQM Q7 UNXdV QM i?2 BMi2`p�HX h?2 7mM+iBQM I(x) BM UNX3V Bb +�HH2/ �M BMi2;`�iBM;
7�+iQ` 7Q` UNXdV- �M/ i?Bb K2i?Q/ 7Q` }M/BM; � ;2M2`�H bQHmiBQM Q7 UNXdV Bb +�HH2/ i?2 K2i?Q/ Q7
BMi2;`�iBM; 7�+iQ`bX
h?2 J2i?Q/ Q7 AMi2;`�iBM; 6�+iQ`b
ai2T R *�H+mH�i2 i?2 BMi2;`�iBM; 7�+iQ`
I = e∫p(x)dx.
ai2T k JmHiBTHv #Qi? bB/2b Q7 UNXdV #v I �M/ 2tT`2bb i?2 `2bmHi �b
d
dx(Iy) = Iq(x)
ai2T j AMi2;`�i2 #Qi? bB/2b Q7 i?2 2[m�iBQM Q#i�BM2/ BM ai2T k �M/ i?2M bQHp2 7Q` yX "2 bm`2 iQ
BM+Hm/2 � +QMbi�Mi Q7 BMi2;`�iBQM BM i?Bb bi2TX
1t�KTH2 NXRj 6BM/ i?2 ;2M2`�H bQHmiBQM Q7
dy
dx− y = e2x
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
R3y
1t�KTH2 NXR9 aQHp2 i?2 BMBiB�H p�Hm2 T`Q#H2K
xdy
dx− y = x, x > 0, y(1) = 2.
1t�KTH2 NXR8 6BM/ i?2 ;2M2`�H bQHmiBQM Q7
dy
dx= xex + y − 1
1t�KTH2 NXRe aQHp2 i?2 BMBiB�H p�Hm2 T`Q#H2K
dy
dx=
x− 1
e2x− 2y, y(0) = 1.
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
R3R
1t�KTH2 NXRd 6BM/ i?2 ;2M2`�H bQHmiBQM Q7
dy
dx=
cosx− y
x, x > 0.
NX8 �TTHB+�iBQMb Q7 .Bz2`2MiB�H 1[m�iBQMb
NX8XR 1tTQM2MiB�H :`Qri? G�r
AM ;2M2`�H- B7 i?2 `�i2 Q7 +?�M;2 Q7 � [m�MiBiv Q rBi? `2bT2+i iQ iBK2 Bb T`QTQ`iBQM�H iQ i?2 �KQmMi Q7
Q T`2b2Mi �M/ Q(0) = Q0- i?2M- r2 Q#i�BM i?2 7QHHQrBM; i?2Q`2K,
1tTQM2MiB�H :`Qri? G�r A7 dQ
dt= rQ �M/ Q(0) = Q0 i?2M Q = Q0ert r?2`2
Ç Q0 Bb �KQmMi Q7 Q �i t = 0
Ç r Bb `2H�iBp2 ;`Qri? `�i2
Ç t Bb iBK2
Ç Q Bb [m�MiBiv �i iBK2 t
A7 r Bb TQbBiBp2- i?Bb #2+QK2b 2tTQM2MiB�H ;`Qri?X A7 r Bb M2;�iBp2- i?Bb #2+QK2b �M 2tTQM2MiB�H
/2+�v T`Q#H2KX
h?2 +QMbi�Mi r BM i?2 2tTQM2MiB�H ;`Qri? H�r Bb +�HH2/ i?2 `2H�iBp2 ;`Qri? `�i2 X A7 i?2 `2H�iBp2
;`Qri? `�i2 Bb r = 0.02- i?2M i?2 [m�MiBiv Q Bb ;`QrBM; �i � `�i2 dQ/dt = 0.02Q Ui?�i Bb kW Q7 i?2
[m�MiBiv Q T2` mMBi Q7 iBK2 tVX LQi2 i?2 /BbiBM+iBQM #2ir22M i?2 `2H�iBp2 ;`Qri? `�i2 r �M/ i?2 `�i2 Q7
;`Qri? dQ/dt Q7 i?2 [m�MiBiv QX _2H�iBp2 ;`Qri? `�i2 Bb yXyk �M/ i?2 `�i2 Q7 ;`Qri? Bb 0.02QX PM+2
r2 FMQr i?�i i?2 `�i2 Q7 ;`Qri? Q7 bQK2i?BM; Bb T`QTQ`iBQM�H iQ i?2 �KQmMi T`2b2Mi- r2 FMQr i?�i Bi
?�b 2tTQM2MiB�H ;`Qri? �M/ r2 +�M mb2 i?2 2tTQM2MiB�H ;`Qri? 7Q`KmH�X
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kykR
R3k
1t�KTH2 NXR3 h?2 rQ`H/ TQTmH�iBQM T�bb2/ R #BHHBQM BM R3y9- k #BHHBQM BM RNkd- j #BHHBQM BM RNey- 9
#BHHBQM BM RNd9- 8 #BHHBQM BM RN3d- �M/ e #BHHBQM BM RNNN- �b BHHmbi`�i2/ BM 6B;m`2 NXRX SQTmH�iBQM ;`Qri?
Qp2` +2`i�BM T2`BQ/b +�M #2 �TT`QtBK�i2/ #v i?2 2tTQM2MiB�H ;`Qri? H�rX
6B;m`2 NXR, qQ`H/ TQTmH�iBQM ;`Qri?
1t�KTH2 NXRN SQTmH�iBQM :`Qri? AM/B� ?�/ � TQTmH�iBQM Q7 �#Qmi RXk #BHHBQM BM kyRyX G2i P `2T@
`2b2Mi i?2 TQTmH�iBQM UBM #BHHBQMbV t v2�`b �7i2` kyRy- �M/ �bbmK2 � ;`Qri? `�i2 Q7 RX8W +QKTQmM/2/
+QMiBMmQmbHvX
U�V 6BM/ �M 2[m�iBQM i?�i `2T`2b2Mib AM/B�Ƕb TQTmH�iBQM ;`Qri? �7i2` kyRy- �bbmKBM; i?�i i?2 RX8W
;`Qri? `�i2 +QMiBMm2bX
U"V q?�i Bb i?2 2biBK�i2/ TQTmH�iBQM UiQ i?2 M2�`2bi i2Mi? Q7 � #BHHBQMV Q7 AM/B�X BM i?2 v2�` kyjy\
�+�/2KB+ v2�` kykR kyeRRR, *�H+mHmb R
R3j
q2 MQr im`M iQ �MQi?2` ivT2 Q7 2tTQM2MiB�H ;`Qri?, `�/BQ�+iBp2 /2+�v X AM RN9e-qBHH�`/ GB##v
Ur?Q H�i2` `2+2Bp2/ � LQ#2H S`Bx2 BM +?2KBbi`vV 7QmM/ i?�i �b HQM; �b � TH�Mi Q` �MBK�H Bb �HBp2-
`�/BQ�+iBp2 +�`#QM@R9 Bb K�BMi�BM2/ �i � +QMbi�Mi H2p2H BM Bib iBbbm2bX PM+2 i?2 TH�Mi Q` �MBK�H Bb
/2�/- ?Qr2p2`- i?2 `�/BQ�+iBp2 +�`#QM@R9 /BKBMBb?2b #v `�/BQ�+iBp2 /2+�v �i � `�i2 T`QTQ`iBQM�H iQ
i?2 �KQmMi T`2b2MiX
dQ
dt= rQ Q(0) = Q0
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U�V dy
dx− 3y = sinx
U#V dy
dx+ xy = x
U+V ydy
dx− x = 1
U/V dy
dx+ xy2 = sin(xy)
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U�V dy
dx+ 2xy = 3x
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1 + y
dy
dx= −x
U+V (1 + x4)dy
dx=
x3
y
U/V y′ + y = sin(ex)
R38
R3e
U2V e−y sinx− y′ cos2 x = 0
U7V dy
dx+ y +
1
1− ex= 0
U;V dy
dx− y2 − y
sinx= 0
U?V dy
dx+ 5y = e−3x
UBV (1 + x2)dy
dx+ xy = 0
UDV y′ − (1 + x)(1 + y2) = 0
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U�V xdy
dx+ y = x, y(1) = 3
U#V y′ =3x2
2y + cos y, y(0) = π
U+V dy
dx=
2x+ 1
2y − 2, y(0) = 1
U/V xdy
dx− y = x2, y(1) = 1
U2V 2dy
dx− y = 4 sin(3x), y(0) = 0
U7V y′ = −4xy2, y(0) = 1
RyX 6BM/ � +m`p2 i?�i b�iBb}2b y′ = −x
y�M/ T�bb2b i?`Qm;? (3, 1)X
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