S `iAA - math.science.cmu.ac.th

S�ì 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


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


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�ìb

eXkXR h?2 S`Q/m+i _mH2 pb AMi2;`�iBQM #v S�ìb

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


RRy

1t�KTH2 eXk 1p�Hm�i2∫

x3HM x dxX

eXkXk _2T2�i2/ AMi2;`�iBQM #v S�ìb

1t�KTH2 eXj 1p�Hm�i2∫

x2ex dxX


RRR

1t�KTH2 eX9 1p�Hm�i2∫

ex cosx dxX


arctanx dxX


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�ì i?2 b2+iBQM #v `2pB2rBM; BKTQì�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


RR9


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


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


1t2`+Bb2 e#


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


RR3

1t�KTH2 eXR9 1p�Hm�i2∫

x3√9− x2

dxX


dx

x2√4x2 − 3

X


1t2`+Bb2 e+


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�ìB�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�ìB�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�ìB�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�ìB�H 7`�+iBQM /2+QKTQbBiBQM +QMi�BMb

i?2 7QHHQrBM; bmK Q7 m T�ìB�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


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�ìB�H 7`�+iBQM /2+QKTQbBiBQM +QMi�BMb i?2 7QHHQrBM; bmK Q7 m T�ìB�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


3x2 + x− 2

(x− 1)(x2 + 1)dxX


Rkk

1t�KTH2 eXRN 1p�Hm�i2∫

x+ 4

x2(x2 + 4)dxX

1t�KTH2 eXky 1p�Hm�i2∫

x3 − 4x

(x2 + 1)2dxX


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.


1t2`+Bb2 e/

q`Bi2 Qmi i?2 7Q`K Q7 i?2 T�ìB�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


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�ìBM; rBi?

k = m �M/ 2M/BM; rBi? k = nX

1t�KTH2 dXR

∑8k=4 k

3 =

∑5k=0(−1)k(2k − 1) =

dXkXk S`QT2ìB2b 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�ìB+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


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.


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 =


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.


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.


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 BKTQì�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�ìBiBQM 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�ìBiBQM 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�ìBiBQM- 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�ìBiBQM- 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�ìBiBQMX 6Q` 2t�KTH2- 6B;m`2 dXd b?Qrb � T�ìBiBQM Q7 i?2


Rjk

BMi2`p�H [0, 6] BMiQ 7Qm` bm#BMi2`p�Hb rBi?

6B;m`2 dXd, T�ìBiBQM 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�ìBiBQMb 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.


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


Rj9

dXjXk S`QT2ìB2b 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


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


Rje

dX9 6mM/�K2Mi�H h?2Q`2K Q7 *�H+mHmb

dX9XR S�ì 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


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


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�ì 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


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�ì 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�ìB+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)


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


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


R9k

dX9X8 AMi2;`�iBQM #v S�ìb 7Q` .2}MBi2 AMi2;`�Hb

1t�KTH2 dXRj 1p�Hm�i21∫

0

arctanx dxX


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�ì 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�ì 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�ì 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�ì 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


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�ìBM; 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


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


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


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


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


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


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


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


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


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


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


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


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


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? p2ìB+�H �bvKTiQi2b rBi?BM i?2 BMi2`p�H Q7 BMi2;`�iBQMX A7 � 7mM+iBQM f ?�b �

p2ìB+�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


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 TQìBQM 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 TQìBQM 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


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


Re8

1t�KTH2 3XRe *QKTmi2∫ +∞

0cosx dxX

1t�KTH2 3XRd *QKTmi2∫ ∞

1

lnx

xdxX

1t�KTH2 3XR3 *QKTmi2∫ 1

−∞

1

3− 2xdxX


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 TQìBQM 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


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


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


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


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�ìB+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�ìB+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�ìB+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


Rdj

1t�KTH2 NXj 6BM/ i?2 T�ìB+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.


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


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.


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.


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.


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).


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


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.


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`QTQìBQM�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`QTQìBQM�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


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` +2ì�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\


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`QTQìBQM�H iQ

i?2 �KQmMi T`2b2MiX

dQ

dt= rQ Q(0) = Q0

h?Bb Bb �MQi?2` 2t�KTH2 Q7 i?2 2tTQM2MiB�H ;`Qri? H�rX h?2 +QMiBMmQmb +QKTQmM/ `�i2 Q7 /2+�v 7Q`

`�/BQ�+iBp2 +�`#QM@R9 Bb yXyyyRkj3- bQ r = −0.0001238- bBM+2 /2+�v BKTHB2b � M2;�iBp2 +QMiBMmQmb

+QKTQmM/ ;`Qri? `�i2X

1t�KTH2 NXky � ?mK�M #QM2 7`�;K2Mi r�b 7QmM/ �i �M �`+?�2QHQ;B+�H bBi2 BM �7`B+�X A7 RyW Q7 i?2

Q`B;BM�H �KQmMi Q7 `�/BQ�+iBp2 +�`#QM@R9 r�b T`2b2Mi- 2biBK�i2 i?2 �;2 Q7 i?2 #QM2X

1t�KTH2 NXkR >�H7@HB72 Bb i?2 iBK2 `2[mB`2/ 7Q` � `�/BQ�+iBp2 2H2K2Mi iQ `2/m+2 Bib [m�MiBiv #v ?�H7X

.2MQi2 #v T i?2 ?�H7@HB72 Q7 � `�/BQ�+iBp2 2H2K2MiX lb2 y = y0e−kt iQ r`Bi2 T BM i2`Kb Q7 i?2 /2+�v

+QMbi�Mi kX A7 i?2 ?�H7@HB72 Q7 `�/BmK@kke Bb Reyy v2�`b- }M/ Bib /2+�v +QMbi�MiX


R39

NXe *QKT�`BbQM Q7 1tTQM2MiB�H :`Qri? S?2MQK2M�

h?2 ;`�T?b �M/ 2[m�iBQMb ;Bp2M BM 6B;m`2 #2HQr +QKT�`2 b2p2`�H rB/2Hv mb2/ ;`Qri? KQ/2HbX h?2b2

KQ/2Hb �`2 /BpB/2/ BMiQ irQ ;`QmTb, mMHBKBi2/ ;`Qri? �M/ HBKBi2/ ;`Qri?X 6QHHQrBM; 2�+? 2[m�iBQM

�M/ ;`�T? Bb � b?Qì U�M/ M2+2bb�`BHv BM+QKTH2i2V HBbi Q7 �`2�b BM r?B+? i?2 KQ/2Hb �`2 mb2/X

6B;m`2 NXk, 1tTQM2MiB�H ;`Qri?

"�`M2ii 2i �HX *�H+mHmb 7Q` "mbBM2bb- 1+QMQKB+b- GB72 a+B2M+2b �M/ aQ+B�H a+B2M+2b URk 2/XV

- S2�`bQM UkyRRVX


1t2`+Bb2 N

RX o2`B7v i?�i #Qi? y1 = ex �M/ y2 = e−x �`2 bQHmiBQMb Q7 i?2 /Bz2`2MiB�H 2[m�iBQM y′′ = yX >Qr

�#Qmi i?2B` HBM2�` +QK#BM�iBQM y = c1ex + c2e−x\

kX aQHp2 i?2 BMBiB�H@p�Hm2 T`Q#H2Kdy

dt= ky, y(0) = y0

7Q` k > 0X

jX *?BM� ?�/ � TQTmH�iBQM Q7 RXjk #BHHBQM BM kyyd Ui 4 yVX G2i P `2T`2b2Mi i?2 TQTmH�iBQM UBM

#BHHBQMbV t v2�`b �7i2` kyyd- �M/ �bbmK2 � +QMiBMmQmb ;`Qri? `�i2 Q7 yXeWX 6BM/ i?2 2biBK�i2/

TQTmH�iBQM 7Q` *?BM� BM i?2 v2�` kyk8X

9X � #QM2 7`QK �M �M+B2Mi iQK# r�b /Bb+Qp2`2/ �M/ r�b 7QmM/ iQ ?�p2 8W Q7 i?2 Q`B;BM�H `�/BQ�+iBp2

+�`#QM T`2b2MiX 1biBK�i2 i?2 �;2 Q7 i?2 #QM2X

8X :Bp2 �M 2t�KTH2 Q7 � }`bi@Q`/2` /Bz2`2MiB�H 2[m�iBQM rBi? mMB[m2 bQHmiBQMX

eX :Bp2 �M 2t�KTH2 Q7 � }`bi@Q`/2` /Bz2`2MiB�H 2[m�iBQM r?B+? ?�b y = e−2x �b � bQHmiBQMX

dX *H�bbB7v i?2 7QHHQrBM; }`bi@Q`/2` /Bz2`2MiB�H 2[m�iBQMb �b b2T�`�#H2- HBM2�`- #Qi?- Q` M2Bi?2`X

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)

3X aQHp2 i?2 7QHHQrBM; /Bz2`2MiB�H 2[m�iBQMbX

U�V dy

dx+ 2xy = 3x

U#V√1 + x2

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

NX aQHp2 i?2 7QHHQrBM; BMBiB�H@p�Hm2 T`Q#H2KbX

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

RRX SQHQMBmK@kRy Bb � `�/BQ�+iBp2 2H2K2Mi rBi? � ?�H7@HB72 Q7 R9y /�vbX �bbmK2 i?�i ky KBHHB;`�Kb

Q7 i?2 2H2K2Mi �`2 TH�+2/ BM � H2�/ +QMi�BM2` �M/ i?�i y(t) Bb i?2 MmK#2` Q7 KBHHB;`�Kb T`2b2Mi

t /�vb H�i2`X

U�V 6BM/ i?2 BMBiB�H@p�Hm2 T`Q#H2K r?Qb2 bQHmiBQM Bb y(t)X

U#V 6BM/ � 7Q`KmH� 7Q` y(t)X

U+V >Qr K�Mv KBHHB;`�Kb rBHH #2 T`2b2Mi �7i2` Ry r22Fb\

U/V >Qr HQM; rBHH Bi i�F2 7Q` dyW Q7 i?2 Q`B;BM�H b�KTH2 iQ /2+�v\

RkX amTTQb2 i?�i 9yW Q7 � +2ì�BM `�/BQ�+iBp2 2H2K2Mi /2+�vb BM 8 v2�`bX

U�V q?�i Bb i?2 ?�H7@HB72 Q7 i?Bb 2H2K2Mi BM v2�`b\

U#V amTTQb2 i?�i � +2ì�BM [m�MiBiv Q7 i?Bb bm#bi�M+2 Bb biQ`2/ BM � +�p2X q?�i T2`+2Mi�;2 Q7 Bi

rBHH `2K�BM �7i2` t v2�`b\


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