Demidovic - Reseni Integrali - Zbirka Zadataka
January 5, 2017 | Author: Dragan | Category: N/A
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1031
5a 2 x 6 dx 5a 2 x 6 dx 5a 2 x77
5 7
a2x7 C 3
2
1033
x4
bx 3
ax 3
1 2
abx 2 C
mn 1 2 x mn 12
a x 4 ax
1 7
abx 4
dt 2p
1
3
1 2 2p 3
t2
3
1 3p
t2
1 3p
3
2px 2
2 3
2 x px C
1
n
x n n 1n nn
x
n1 n n1 n
nx u
1
1
1 n
u 1nn du
1n
n
n
n
1 u n n n 1n n
1
1 un n 1 n
3
3
3
3
dx x 2 7
1 7
arctan
2
a 2 x 3a
dx x 2 10
1 10
dx 4x 2
1 7
1
2 5 a
a 5
a 2 5 x2 5 2
1 x
3
3
a 2 4a 2 6 x a 4x a
1
3
4
2
2
x2 C
5 x3 5 3
4 3
3a
2 3
7 x3 7 3
x3 3
a2x
9 5
4 3
5 3
a x
1 2
dx 8x 2
arcsinh
1 4
2x 2 2x 2 4x 2 1 2
dx 2x
9 7
2 3
a x
1 3
x3 C
2xC
5 2
x 1x x 1dx x 1 dx x x C 2 5
dx 3 7
7 3
2
x3
dx x
x 63 x C
10 3
4 3
x 2x
23
dx
dx
sinh 2 cosh 2
cos 2 x sin 2 x
dx
1 2x 2
dx
1 2x 2
dx
xC
1cos 2 x cos 2 x
dx
1 cos 2 x
dx dx
1 cos x
sin x x C
1cosh 2 x cosh 2
dx cosh 2
dx
sinh x cosh x
x tanh x x C
dx
1sin 2 x sin 2 x
dx
dx sin 2 x
dx cot x x C
1049b
coth 2 xdx x 4 x 2 2
2x 2 2x 2
1049a
1040
xC
2x 2 2x 2 arcsinh 12 2
sin 2 x cos 2 x
tanh 2 xdx
7 3
cot 2 xdx 3 2
3 x2
xC
1048*b
4
1039
x 2 1x 2 2
1 10
3 13
x
13 3
3 7
7 3
x 2 3x
1 3
cosh 2 x sinh 2 x
1050
3 x e x dx 3e x dx
dx
1sinh 2 x sinh 2 x
dx
dx sinh 2 x
3
a 2 2a 2 x 4a 2 x 4ax 2 2x
xC
arctanh
arcsinh
tan 2 xdx
a 3 x 3 3 dx a 2 3a 3 x 3 3a 3 x 3 x 2 dx
x
2
a 2 4a 2 x 6a 23 x 2 4 x2
3
1048*a
1
1
u n nx n C
1038
13 3
1
3
1 1 x 2 1 1 2
arcsin
ndx du
3 13
C
1047
nx 1nn dx
2
2n 1 1 2
2n 1 1 2
1046
n1
n xn1n C
1037
du n
x
1045 1
t 2 dt
1 2p
x n dx
u 1nn
2
1043
b2x6 C
1036 dx x
mn 1 1
2 x 2 mn 1 1
C
2a 2 x 4ax 4 a x 2 2x 2
2pdx dt
n
2m 1 1
2 x 2m 12 1
1044
2px t| t
1
2 dx ax 2 a 2 a x x dx ax 2 a 2 4a 2 x 6ax 4 a x 2
3
1 2
1041
1
2n 1 x 2 2n 12
1
1042
1
2px dx
2
a 2
1035
2m 1 2 x 2m 12
1
a bx 3 2 dx a 2 2abx 3 b 2 x 6 dx a 2 x 2ab x44 b 2 x76
1
1
dx x 2m x 2 2x mn x 2 x 2n x 2 dx
a 2 x 2 a 2 4a 2 x 6ax 4 a x 2 x 2
1034 a2x
xx ax bdx xx 2 xb ax ab x 3 x 2 b ax 2 xabdx 4 3 3 2 x 3 dx b x 2 dx a x 2 dx ab xdx x4 b x3 a x3 ab x2 1 3
x 2m 2x mn x 2n x
1
6x 2 8x 3dx 6 x 2 dx 8 xdx 3 dx 6 x3 8 x2 3x 2x 3 4x 2 3x C
1 3
dx
x 2m 2 2x mn 2 x 2n 2 dx
1032
1 4
x m x n 2 x
dx coth x x C
x1
3e x u x x
3 e 1 ln 3dx du 3e x
du 3 x e x 1ln 3
du 1ln 3
1 1ln 3
3e x 1ln 3
u
2 xdx dx 2
C
a
adx ax
dx ax
a x u| a
du u
a ln u a lna x C
dx
dx dx
2x12 2x1
2 2x1
dx x
2 2x1
dx
x ln u x ln2x 1 C
xdx bxdx 1b abx abx dx x ab abx b
1 b
bxaa abx
1 b
dx
a b
du b
bxaa abx
1 b
dx
dx
a b
dx abx
a
x b
a b2
du u
x b
a b2
ln u
x b
a b2
5 dx 15
dx dx
x4 x1
dx
b a
x axb xdx b a x dx a x x dx x x x dx dx a a dx b dx b x x x x a dx dx a a dx b x ax dx x x a ba dx dx ax ax b x x
ba
b xa
du
1x
2
ba xa
dx
b2 xa 2
u
b2 t
ax
ba 2
du u
xa 2 2ab lnx a
b
dx x
x x1 2
dx
x11 x1 2
dx
dx x1
dx x1 2
x 2 11 x1
dx lnx 1
1
ax
ba 2
b2 xa
x2 2
C
lnx 1
dx x1 2
x 1 t| dx dt dt t2
lnx 1
1 t
lnx 1
ln u a x
ba 2
lnx C
bdy 1y
b
dy 1y
1 y t| 2 x1
dx 7 lnx 3
x
x4 4
dx a 2 x 2ab lnx a b 2
dt t2
1061
x33 x3
dx a 2 dx 2
lnx 1
x 1 : x 1 x 1 x2 x
1060*
1056 x 2 1 x1 2
dx 5
7 lnx 3
a x 2ab lnx a b 2
ax
x 2 99 x3
4 11 dx x2 dx x1 x x1 dx x1 x 2 1 lnx 1 x1 dx lnx 1 lnx x 2 1x1x1 x1x1 dx dx x1 x1 2
xa 2 2ab lnx a
dx x3
3 5x 15 lnx 3 7 lnx 3 3 C
2
dx du
dx x3
dx x3
dx dt
lna bx C
x u|
dx 7
x3 3
x2 2
1059
1055
x x3
x a t|
u
dx 5
2
bdx du x b
x2 x3
3 lnx 1 x2 x x 3 x 2 x 1dx 3 lnx 1 14 x 4 13 x 3 x 2 2x 3 lnx 1 C
a bx u|
x 2 lnx 1 C
3 lnx 1 x 1dx x 1x 1dx
1054
x 4 x 2 1 x1 x 4 1 x1
3 lnx 1
2dx du du u
dx
x2 3x 9 lnx 2 1 2 x 2x lnx 2
2x 1 u| x
x2 2
1058
1052** 2x3 2x1
x 2 5x7 x3
x 3dx 9
dx du
1057
1051**
dx x1
dy dt b
dt t
2b t 2b 1 y C
1062
a bx dx
1 x1
C
dx xa 2
x
a bx t|
1071
bdx dt t
1b t dt
dt b
2 3b
3 2
t
3 2
2 3b
a bx C
x
dx
x 2 1
dx 75x 2
x 2 1 t|
1073
2xdx dt
dt 2x
x t
1 2
dt t
x2 1 C
t
2x5 3x 2 2
dx x
dx
1 x
dx
1 x
ln xdx
dt 1 2
2 x
ln 2 x C
1 3
dx x 2 53
1 3
15 5
15 5
arctan
15 15
x
15 5
arctan
1 7
dx x 2 87
1 7
1 4
14 arctanh
x 2
1 ab
1 4
14 28
x 14
14 4
arctanh
xC
ab ab ab
1 ab
arctanh
dx ab ab
x ab
ab
ab
ab
1
arctanh
ab ab
1 a 2 b 2
arctanh
x ab ab
x a 2 b 2 ab
x2 x 2 2
dx
x3 a 2 x 2
a3
x 2 22 x 2 2
dx dx 2
dx x 2 2
x 2 arctan
1 a
a 3 1a
dx
x 3 axax
axa 2 axx 2 axax
dx
dx
x 3 a 3 a 3 dx axax x 3 1 a a arctanh a
x
5 2
1 2 2
2
5 5
arcsin
1 3
2 14 7
arcsinh
2
2 4
x
arcsinh
2 14 7
xC
dx 7 5
x 2
2x5 x 2 23
dx
35 7
xC
dx 5
2x x 2 23
dx x 2 23
dt t
6 2
5
2
2 3
dx
1 5
7 5
t|
lnx
6 2
arctanh 5 6
1 2
6 arctanh
1 3
x
ln t
5 6
6 arctanh
1 2
x 6
x 6 C
32x x 2 75
dx
3 5
dt t
dx x 2 75
1 5
2xdx x 2 75
3 35 35
arctan
35 7
x
35 7 35 7
arctan arctan
x
1 5
x
1 5
3 35 35
lnx 2
7 5
dx
3 5 5
35 7
arctan
x
1 5
ln t
C
a 2 axx 2 ax 2
arctanh
x a
arctanh
x a
a lna x xdx
dx a 3
1 a
2 2
1 2
x a
2
a2 ax
2
dx
a3
1 a
dx
a3 a 2 x 2
arctanh xxa ax 2
3 5 5
3 5 10
x 2 4
2
dx
x 2 5x42 x 2 4
lnx 4 arctan
1 2
dx dx 5 xC
xdx x 2 4
x a
x3 x 2 4
dx x 2 4
3x1 x 2 15
x x 2 15
5 5
dx
dx x 2 15
t|
dt 2
t
2 t
5 5
3 5 10
arcsinh 5 x 5 5
arcsinh 5 x
dt t
5 5
arcsinh 5 x
1 5
3 5 5
x
3 ln x x 2 4
2
5 5
arcsinh x 5 C
dx
xdx x 2 4
3
dx x 2 4
x 2 4 t|
dx
x a lna x a arctanh
2
1 5
5 5
1076
dx
x a
2xdx dt
C
x 2 5x6
dx
x2
xC
a 3 x 3 axax
ax
arctanh
3x1 5x 2 1
2xdx dt
a 2 axx 2
1070
t|
3 35 35 3 35 35
C
1069
1075
dx ababx 2
1068
x 2
2xdx dt
1067
2 3
x2
dx 7x 2 8
5 5
1 3
32x 5x 2 7
xC
1066
dx 7 8
1074 1 2 t 2
1065 dx 3x 2 5
1 3 1 3
2 x tdt 2 x
2xdx dt
x ln x x
ln x t|
1 8
dx
x2
1064
1072
1063**
dx 78x 2
1 2
1077
dt 2
t
3 ln x x 2 4
2 t 3 ln x x 2 4
1 2
dt t
x 2 4 3 ln x x 2 4
C
1 5
2xdx x 2 75
xdx x 2 5
1
x6 t2| 2
x 2 5 t|
x 3 t|
2xdx dt
3x 2 dx dt
dt 2
1 2
t
dt t
1 2
1 2
ln t
1078
dt 3
lnx 2 5 C
t 2 1
1 3
dt
arcsin x 1x 2
dt 4
1 4
t
dt t
1 4
1 4
ln t
axb a 2 x 2 b 2
1 a
dx a 0 ax
b2
ax
t|
a
b a2
dx
2 ax 2 ba
2
1 a
axb 2
ax 2 ba
1 a
dx
ax 2
ax 2 ba
dx
b a2
1 a 1 2a
dx 1x 2
dt 2
b a2 b2 a
dx
t
lnax 2
2dx x 2 4
a
dx 2 x2 b2 a
a b
arctan x 1 a
arctan
a b a b
1 2a
dt t
1 a
arctan
a b
x
1086
2xdx dt
dt
a 4 t 2
1 2
t
arctan
a 4 t 2
1 2
arctan
x2 a 4 x 4
du 8
u
t
1t 2
1082 x 2 dx x 6 1
1 3
C
x 14x 2
1 2
arctan 2x 14x 2
dx
dx
t 2dx 14x 2
dt 2
1 8
ln u
3 2
dt 1 2
2 3
3
t2
1 8
ln u
1 3
3
t2
arctan 2x C
lnx 1 x 2 u|
C
dx
du u
du
2 u 2 lnx 1 x 2 C
1087
ae mx dx
3x 2 dx dt dt 3
x 2
dx
dx x 3 t|
1 3
x 2 1
x6 t2|
arctan 2
1x 2 lnx 1x 2
1081
dx
ln1 4x 2
x 2 t|
x2 1x 6
1 4
8xdx du 1 8
1 2
u2 4
1 4x 2 u|
x4 t2| 2
3
u2
du
x arctan 2x 14x 2
xC
a 4 x 4
dt 2
2 3
u|
udu
1 2
a 4 t 2
u du
du
1085
xdx
x 2
arctan 2
1
ln x 3 x 6 1
x 2 4
dx
1080
1 3
x 2
arctan
x2 b2
2axdx dt
1084
ln2x 2 3 C
1079
ln t t 2 1
arcsin x u|
dx
4xdx dt
1 3
C
1083
xdx 2x 2 3
2x 2 3 t
t 2 1
mx u| dt 1t 2
1 3
arctan t
1 3
mdx du
arctan x 3 C
a u ae u du m m e du
1088
4 23x dx
a m
eu
a m
e mx C
2 3
3
arcsin x 2 C
2 3x t| 3dx dt 4t
dt 3
13 4 t
4 t u|
1095
4 ln 4dt du
t
13 4 t dt du
4 t ln 4
3 ln1 4 du 3 ln1 4 u
4t 3 ln 4
1
ex x2
1 x dx x2
3 ln1 4 4 23x C
1089
1
1096
5
1090
e ax e ax 2 dx e 2 ax dx 2 dx 2dx a
x dx x
x t|
dx x e2 a
dx 2 x
2 ax t|
dt
5 t 2dt 2 5 t dt
dt
e t a2 dt 2x a2 e t dt a2 e t dt 2x x x 12 ae 2 a 2x 12 ae 2 a C
a 2
2 ln 5
2 ln 5
5t
5
x
C
1097
e t dt
ex e x 1
dx
e x 1 t|
1091 a x b x 2 axbx
a b
dx x
dx
a 2x 2a x b x b 2x dx axbx x 2x ba dx
ax bx a b
ln
a b
dx 2dx x
2x
x
b a
ln
b a
bx ax
C
1
1
3
3 x u| 2 3 dx du 2 2 3 ln a
1
2 ln a
ln t lne x 1 C
a be x t| be x dx dt
dx2 dt at
dt t
e x a be x dx
12 x t|
2 ln a
au
1098
dx a 2x 2 x dx a 2 x dx a 2 x dx a 2 x dx a u 23 du a t 2dt
a 2x 1 ax
e x dx dt
dx
1092
dt
e dt e t e x C
t
t
t|
t
e t e t dt 2 e e2 dt 2 sinh tdt 2 cosh t C e t e t dt e t dt e t dt e t e t C
dx
3x 2
t 1 2
a 3 a x C
1093
dt b
1b t dt
2 3b
3
t2
2 3b
3
a be x 2 C
1099 1
e ax 1 3 e ax dx
e x 2 1 xdx e x 2 1 xdx
x
e a 1 t|
x 2 1 t|
1 a
2xdx dt
3 4
4
at 3
3 4
4
x
ae a 1 3 C
1100*
1094
dx 2 x 3
2 x 3 t|
x7 x 2 dx
2 x ln 2dx dt
2
x t|
2xdx dt dt 2
1
t adt a t 3 dt
dt e t 2 12 e t dt 12 e t 1 x 2 1 2e C
7t
x
e a dx dt
1 3
1 2
7 t dt
1 2 ln 7
7t
1 2 ln 7
2
7x C
dt 2 x ln 2
1 ln 2
t
dt 2xt
1 ln 2
1
arctanh
9 4
t 32 9 4
1 ln 2
dt t3t
2 3 ln 2
1 ln 2
arctanh
2 3
dt t 2 3t
t1
1 ln 2
dt t 32 2 94
2 3 ln 2
2 3
arctanh
2 x 3 1 C
1107
1101
cos x
a x dx 1a 2x
x z ax u
dx 2 x
a x ln adx du
du ln a1u 2
1 ln a
du 1u 2
1 ln a
1 ln a
arctan u
arctan a x C
1102
e bx 1e 2bx
1108
sinlg x dxx sin lnln 2x dxx
dx
ln x ln 2 dx x ln 2
be bx dx du du b
1b
1u 2
du 1u 2
1 b
arctanh u
1 b
arctanh e bx C
1103
dz
cos z2dz 2 cos zdz 2 sin z 2 sin x C
e bx u
dx x
z| dz
sinz ln 2dz ln 2 sin zdz ln x ln 2 cos lg x C ln 2 cos z ln 2 cos ln 2 1109
et 1e 2t
dt
sin 2 xdx
t
e u
2x z|
2dx dz
du
arcsin u arcsin e t C
1u 2
x 2
1 2
1 2
dx
1 2
cos 2xdx
e t dt du
1cos 2x 2
cos z 12 dz
x 2
1 4
cos zdz
1 2
x
1 4
sin z
1 2
dx
1 2
cos 2xdx
1 4
sin z
x
1 4
sin 2x C
1110*
1104
cos 2 xdx
sina bxdx
1 2
1 2
x
1 b
sin zdz
1 b
cos z
1 b
cosa bx C
1 2
x
1 2
cos z dz2
1 2
1 4
x
cos zdz
1 2
x
1 2
x
1 4
cos
x 2 dx 2
cos 2xdx
sin 2x C
1111
1105 x 2
1 2
2dx dz
bdx dz sin z
1 cos 2x 2 dx
2x z|
a bx z dz b
1 2
sec 2 ax bdx
dx
dx cos 2 axb
ax b t
z
adx dt
dz
cos z 2 dz
2 cos zdz
2 sin z
2 sin
x 2
C
1106
cos ax sin ax dx sin ax 2 sin ax cos ax cos axdx 1 2 sin ax cos axdx 1 sin 2axdx x sin 2axdx 2
2
2
2ax z
cos 2 t
1 2a
sin zdz x
1 2a
cos z x
1 2a
cos 2ax C
1 a
dt cos 2 t
1 a
tan t
1 a
tanax b C
1112
cot 2 axdx
cos 2 ax sin 2 ax
dx
1sin 2 ax sin 2 ax
dx
dx sin 2 ax
dx
ax t adx dt
2adx dz dz x sin z 2a x
dt a
1113
dt a
sin 2 t
x
1 a
dt sin 2 t
x
1 a
cot t x
1 a
cotax x C
dx sin ax
1119
x a dx a
t
tan xdx
dt a
adt sin t
dt sin t
a ln tan
t 2
a ln tan
x a
a lntan
2
x 2a
C
4
t
dt 5
dt 1 t 151 cos t 15 ln tan 2 4 3 cos t 1 1 1 1 lntan 2 5x 4 4 15 ln 15
tan
5 2
1 8
x
C
sin 2x 1cos 2x
dt a
sin t
1cos 2x
dx
dx
cos x sin x
dx
1 2 1 2
1cos 2x 1cos 2x
dx
1cos 2x 1cos 2x
dx
1cos 2x
1cos 2x
1cos 2x
1cos 2x
dx
1cos 2 2x 1cos 2x
dx
t
dt sin t
1 a
t 2
ln tan
1 a
ln tan
axb 2
cot
C
2xdx dt
dt cos 2 t
1 2
tan t
1 2
2
tan x C
2xdx dt
dx
u| du
sin u t|
sin t dt2 12 sin tdt
1 2
cos t
1 2
cos1 x 2 C
5 2
dx
1 sin 2 x 2
2 sin x 2
1 dx x 2
dx sin x 2
dt t
tan x
5 ln t 5 lnsin u 5 ln sin dx x
x u|
2 dx dt
x 2 ln tan
du
1123
dx sin 2 x 2
x 2 t
cos u sin u
cos udu dt
1118 1
x 5
cot u5du 5
1x t
sin t
du
cot
x 5 dx 5
2
dt 2
ln1 cos 2x C
u|
dx tan 5x
x sin1 x 2 dx
x 2
dx
1 2
1122
1 2
1117
1 sin x 2
x ab
ln t
cot ua bdu a b cot udu a b lnsin u a b ln sin
x2 t
cos 2 t
1 2
x ab dx ab
xdx cos 2 x 2
dt 2
1121
1 a
dt 2
1116
1cos 2x
1cos 2x
2 sin 2xdx dt
adx dt
dx
sin 2x 1cos 2x
1cos 2x
1 cos 2x t|
ax b t
sin 2 2x 1cos 2x
dx
dx 12 ln t 12 ln1 cos 2x C
cot xdx
dx sinaxb
1cos 2x
1120
1115
dt2 t
5dx dt
dx
1cos 2x
2 sin 2xdx dt
dx 3 cos5x 4
5x
dx
1 cos 2x t|
1114
1cos 2 2x 1cos 2x
sin x cos x
dt 2 sin 2 t
x 2 2
x 2
2 2
dt sin t
cot x 2 C
2 2
dt sin 2 t
x 2 ln tan
t 2
2 2
cot t
dx 2 x
du
2 tan udu 2 tan u 2
sin u cos u
du
x 5
C
x ab
C
cos u t|
ax t
sin udu dt
adx dt
2
dt t
2 ln t 2 lncos u 2 ln cos x C
1 a
1124
x cotx 1dx
cos tdt dz 1 a
2
x 1 u 2xdx du
dz z5
cot udu
1 2
1 2
lnsin u
1 2
ln sinx 2 1 C
sin 3x 3cos 3x
1125
1 4a sin 4 t
1 4a sin 4 ax
C
dx
3 cos 3x z|
dx sin x cos x
dx sin 2x 2
2
dx sin 2x
3 sin 3xdx dz
2x u 2
du 2
sin u
sin 3x dz z 3 sin 3x
13
dz z
1 3
ln z
1 3
ln3 cos 3x C
1130
2dx du du sin u
ln tan
u 2
lntan x C
sin 2x 2
dx
sin x cos x cos 2 xsin 2 x
1126
1 2
dx
cos 2x
sin 2x cos 2x
dx
cos 2x t|
cos sin dx x 2
x 2 dx 2
x 2
2 sin 2xdx dt
u
du
1131
2 cos u sin udu
1 2
C
1 2
cos 2x C
1 3t 2 z 6tdt dz
6x t
6dx dt sin 3 t cos t dt6
1 6
2 6
tan 3 3x sec 2 3x dx cos
cos tdt dz z4
1 24
sin t 4
3
3
3
z dz 29 z 2 29 1 3t 2 2 29 1 3 cos 2 x 2 C
1132
sin 3 t cos tdt
sin t z 1 24
sin 4 6x C
1 3
3
1128 dx
t
2 t 1 3t 2 dt
sin 3 6x cos 6xdx
cos ax sin 5 ax
1 2
sin xdx dt x 2
1127
1 24
dt t
cos x t|
cos udu dz
z 3 dz
14
t
1 3 cos 2 x sin 2xdx 2 sin x cos x 1 3 cos 2 x dx
2 zdz z 2 sin 2 u sin 2
1 6
dt 2
sin u z
1 4az 4
1129
cot u du2
dt
sin t z 2
cos t sin 5 t
3
1 4t 4
3
1133
x 3
sin
1 2t 2
x 3 x 3
1 cos 2
x 3
dx
sin 3 13 x cos 5 13 x
dx
t| x 3
dt
sin 2 3x dt t5
sin 3 cos 3
3
3 4 cos 4
x 3
1t 2 t5 3 2 cos 2
dt 3 x 3
C
1 t5
dt 3
1 t3
dt
tan x cos 2 x
dx
1138
2 sinh 5x 3 cosh 5xdx 2 sinh 5xdx 3 cosh 5xdx
tan x t dx cos 2 x
dt
t dt
2 3
5x t t
3 2
5dx dt
3 2
2 3
tan x C
2 5
1134 2
a
2 cos 3 x 2 sin 3 x sin 2 x
dx
cot 3 x sin 2 x
dx
8 3
sin x
sh 2 xdx e e2 2 dx
dx
x
cos xdx dt
3 5
5 3
3t
dt 8 3
sin x
3 5
3 5
dt
5 3
sin x sin x
35
5
3
t 5 sin 3 x
5
2 cot 3 x sin 2 x
b
cos x
3 5
2 3
1 sin x
3 5
dt t sin x
3 5
dt tt
3 5
3 5
dt t
8 5
C
t
5 3
3 5
5 3
cot x C
dx cos 2 3x
3dx dt
3 sin 3xdx dz
dt sin t
dz 3 z2
dx
2 a
1 3
dt cos 2 t
1 3
z 2 dz
1 3
cot t
1 3z
1 3
cot 3x
12 sin ax cos ax sin ax
dx
dx sin ax
cos tdt
1 a
csc 2 3x ba cot 3x
dx
1 3 cos 3x
C
2 cos axdx
bat
dx
1 4
dt 2
et
1 2
x
1 8e t
1 8
e 2x
1 2
x
1 8e 2x
C
2 e x e x
dx
dx
2 e 2x 1 ex
2e x e 2x 1
dx
2 arctanh t 2 arctanh e x C
ln tan
t 2
2 a
sin t
1 a
ln tan
ax 2
dt t 2 1
2 e x e x
dx 2
dx e 2x 1 ex
2 a
sin ax C
dx 1 2
cosh
x 2
1
lnb at
lnb a cot 3x C
1144
dt t
x 2
1
dx 1 4
x
sinh x cosh x
dx
cosh x t
cosh
cosh 2 2x 1
2 arctanhe u 2 arctanhe 2 C
sinh xdx dt 1 3a
1 2
du
du sinh u
1143
ba cot 3x
1 3a
e x dx e 2x 1
u
tanh xdx
dx sin 2 3x
dt bat
2
2 arctan t 2 arctan e x C
dx sinh x cosh x x 2 dx 2
3dx sin 2 x dt 1 3
et
1 4e 2x
1142
cot 3x t dt3
2dt t 2 1
2
1137
1 8
1 2
e dx dt
adx dt 1 a
dx cosh x
ax t
e 2x
x
1136 cos axsin ax 2 sin ax
1 4
ex t
cos 3x z
cos 2 t
sinh 5x C
1141
sin 3x cos 2 3x
3x t dt 3
x
e t dt2 12 x
dx sinh x
3 5
3 5
cosh 5x
e x dx dt
1135
2 5
ex t
t dt 1sin 3x cos 2 3x
sinh t
1140
2 3
sindx2 x dt
1 4
dx
cot x t|
3 5
cosh t
2dx dt
2 3
dt
2 5
2x t
sin x t| 2 cos 3 x 8 sin 3 x
cosh tdt
3 5
1139
2
cos 3 x
5 3
5 3
sinh tdt
ln t ln cosh t C
1 2
dx sinh
x 2
2
dx sinh 2x
coth xdx
cosh x sinh x
dx
13 3 ln t t 2 2
sinh x t
cosh xdx dt
dt t
5 x2 t 2xdx dt 12 5 t dt 125 t 5 6
5 12
1 2
dx
dt 4
1 4
t
1 4
ln t
lnx 4 4x 1 C
1147 dx x4 t 4x 3 dx dt
dt 4 t 2 5
1 4
x 3 1 x1 3
1 3
dt t 2 5
5 20
arctan
t 5 5
5 20
arctan
x4
5
x 6
1 3
3 3
3 arcsinh
1 2
1 2
dx x 2 23
1 2
eu
1 2
2
e x C
1sin x xcos x
2 3x t
1 sin xdx du
2
3xdx tdt 1 3
t 2 z
dx
x cos x u
dx 2
3t tdt t 2 3x
1
e 2 x dx
1152
3 23x 2
x 2 x 2 lnx 1 C
1
du 12 e u du
23x 2
1 2
2
tdt zdz
3t t2
t 2 2 3
tdt
3 3
3t t t 2 2
dt
3 3
3 t t 2 2
3 3
t t t 2 2
3
1 t t 2 2
du u
ln u lnx cos x C
1153
tan 3xcot 3x sin 3x
dx
3x u 3dx du
2
3 ln t t 2 2 2
dx x1
3
1 2
3 3
1 2
2 2
2 3x 2 2
2 3xC dx x 2 23
1 z 2 2z
zdz
3 2 arctan
t2 2
3 2 arctan
x 6
x 6 C
2 x1
1 2
1 3
3 2 arctan
arcsinh
2 e u du 2e u 2e 2 x C
1149
2
x 6
dx 23x 2
dx2 du
2xdx du
dx
12 x u
x 2 u
1 2
3 23x 2
1 2
dx x 2 dx xdx dx 2
x3
dx ex
xe x 2 dx
eu 2
dz
3 2 arctan
1151
C
5
1148
1 2
x 1 : x 1 x 2 x 1 x3 x2 1 x 2 x 2 x 1 x x1 2
4x 3 4dx dt
x3 x 8 5
dx
1 2
2 3x 2 3 x
6 arctan
6 arctan
3
1 z 2 2
13 3 ln t t 2 2
1150
x 3 1
x 4 4x1
x 4 4x 1 t
1 2
1 zdz tz t
2 3x 2 2 3x 2 2
3 ln
3 23x 2 23x 2
6
5 x 2 5 C
1146
1 3
x 5 5 x 2 dx
3 ln t t 2
13 3 ln
1145
dt 2
2
13 3 ln t t 2 2
ln t ln sinh x C
5 t
1 3
3
1 2
1 3
tan ucot u sin u
sin u cos u
u cos sin u sin u
1 3
du
1 3
du
1 cos u
1 3
cos u sin 2 u
xdx 1x 4
sin u t
x 2 u|
cos udu dt 1 3
ln
2xdx du
u dt 1 1 ln 1sin cos u 3 3 t2 1sin 3x 1 3 sin 3x C cos 3x
1 3
ln
1sin u cos u
1 3t
1 3
ln
1sin u cos u
1 3 sin u
dx x dt t2
1 t
1 ln x
C
1 a
tan 2 x2
dx cos 2 x
dx
dx cos 2 x
sin 2 u cos 2 u
du
1 a
1cos 2 u cos 2 u
du
1 a
du cos 2 u
1 a
du
1 a
tan u
u 2
ax C
sin 2 2x dx
tan 2 x2
x 2 dx 2
dt
dt
ln t t 2 2
t 2 2
ln tan x tan 2 x 2
2
dx 2x 2 1
2 2x 2 1
dx
x 2 2x 2 1
dx
sec 2 xdx 4tan 2 x
tan x t
2x 2 1 t
dx cos 2 x
4xdx dt dx
1 cos 2t 1 cos 2t t
1 2
1162
x 2x 2 1
x 2 12
t dt
2 sin 2 tdt 2
C
1156
dt 4t 2
2 arctan x 2
1 4t
1157
2 arctan x 2
1 42x 2 1
C
a t dt
1 ln a
at
4t 2
x a dx a 1 ln a
a sin x C
arcsin
t 2
arcsin
tan x 2
C
dx cos ax
sin x t cos xdx dt
dt
dt
1163
a sin x cos xdx
a
t dt
dt cos t
a lnsec t tan t a ln
1sin ax cos ax
1164
1158 x2 3 x 3 1
dx
3
dx x
3x 2 dx du du 33u
1 2
3
1ln x x
dx
1 ln x t
x 3 1 u|
1159
1 2
tan ax
1 a
1161 sec 2 x
tan 2 udu
1 a
tan x t
arcsin x 2 C
adx du
dt
1155
1 2
ax u|
ln x t
arcsin u
tan 2 axdx
dx x ln 2 x
1 2
1160
1154
du 2 1u 2
u2
1 2
3
x 3 1 2 C
dt
t dt 3
1165
3 4
3
t4
3 4
3
1 ln x 4 C
C
1 2
sin 2t
x 2
1 2
sin x C
tan x 1
dx x1
x1 t dx 2 x1
dt
2 tan tdt 2 lncos t 2 ln cos x 1
C
1166
dt sin t
t ln 1cos sin t
1 2
1 2
1cos x 2 sin x 2
ln
1x 2
dx 1x 2
dx
e arctan x 1x 2
dx
x ln1x 2 1x 2
dx
dx 1x 2
ln1 x 2 t 2xdx 1x 2
du
1 2 t 4
2t 1t 2
cos x
1t 2 1t 2 2dt 1t 2
dx 2t 1t 2
1t 2 1t 2
2t 1t 2 1t 2 1t 2 t 2 2t1
2dt 1t 2
2t1t 2 1t 2
1sin 2x cos 2x
dx
tan 2xdx
dx cos 2x
1 2
lncos t
dx 2
2
dx t
dt 1sin t 2 sin t
2
1sin t 2 sin t
2
t 2 ln 1cos 2t cos t sin t
2
2 ln
12 sin tsin 2 t sin t 1cos x 2
sin
x 2
2
dt 2
x 2
dt sin t
cos
2 dt sin t x 2
C
x2 x 2 2
dx
x 2 22 x 2 2
dx dx 2
dx x 2 2
x 2 arctanh
1 2
x 2 C
1171
2
t 2 2t1 2t1t 2 1t 2
atb 2t1t 2
dt
dt
t1 2t1t 2
dt
1x 2 dx x1x 2 x dx 1x 2
ctd 1t 2 2
x 2 2x1 x1x 2
dx
2 arctan x
x 1x 2 dx x1x 2
dx 2
dx 1x 2
dx x1x 2
1 x2 t 2xdx dt
dt 2
t
1 x1x 2
ac 0 b 2c d 1 a c 2d 2 b d 1 a 1, d 0, c 1, b 1 t 2 2t1 2t1t 2 1t 2
x 2 x 2
1170
t 2 2t 1 at b1 t ct d2t 1 t 2 t 2 2t 1 at at 3 b bt 2 2ct 2 ct ct 3 2dt d dt 2 t 2 2t 1 t 3 a c t 2 b 2c d ta c 2d b d
sin
arctan x
dx
sin x
1sin
x 2
1168 sin xcos x sin xcos x
dt
e u du 12 tdt arctan x e u 14 ln 2 1 x 2 arctan x C
dx
1 2
arctan x
1sin 2x cos 2x
1169
e arctan x x ln1x 2 1
arctan x u
e
dx
12 sin x cos x sin 2 xcos 2 x
dt 1sin t 1 1 12 cos t 2 tan tdt 2 ln cos t 1sin 2x 1 ln cos 2x 2 lncos 2x C
C
1167
2tdt du
2dx dt
2xdx dt 1 2
21 tdt dz
2x t
x2 t
1 t2 u
2 dz du 22 ln z 22 ln u 2z 2u 2 ln2t 1 t ln1 t 2 ln2 tan 2x 1 tan 2 2x ln1 tan 2 2x C 2 xcos x xcos x sin xcos x x cos xcos 2 x sin dx sin dx sin x2sinsin dx 2 xcos 2 x sin xcos x sin xcos x sin xcos x
xdx sin x 2
2t 1 t 2 z
t 1t 2
dt
2 arctan x
a x
dx x1x 2
1 2
ln t 2 arctan x
bxc 1x 2
1 a1 x 2 xbx c a ax 2 bx 2 xc 1 x 2 a b xc a ab 0 a 1b 1c 0 bxc x 1x I 1 ax dx 1x 2 dx a ln x 2 dx ln x I 1172
1 2
dx x1x 2
2
ln1 x 2 arctan x ln x
1 2
2
1 2
ln1 x 2
ln1 x 2 arctan x ln x C
e sin 2 x sin 2xdx 2 e sin 2 x sin x cos xdx 2
sin x t e dt e
sin 2 x
C
dx
53x 43x 2
5 43x 2
dx
dx
3x 43x 2
: 3
5
43x 2 : 3
dx
3x 43x 2
2t T 2 T
5 3
3
dx 4 3
x 2
1 2
3 arcsin
dt 6
5 3
t
1 2
arcsin
3x t
5 3
3x
3 arcsin
1 2
1 2
3 x 4 3x 2 C
dt ext
t
dx x
1 2
dt t 2 t
1cos 2ax sin 2ax
C
T 2
cos x
cos 2t 0 C T
dt
xdt x4t 2
arccos
x 2
dt 4t 2
1 4
ln
2t 2t
1 4
ln
ln x2 2ln x
C
dx
1x 2
dt t 12
2
14
u
du u 2 14
1
1 4
2
ln
u u
1 2 1 2
ln
u u
1 2 1 2
ln2u 1 ln2u 1
1 2
ln2t 1 ln2t 1 ln2t 2 ln 2t ln2e x 1 2 ln 2e x 1 C 1175 dx ababx 2
1 ab
dx 1x 2
1 ab
e 2x 1
1 2
dx dt
arccos 2
x 2
C
e tan x sec 2 xdx e tan x cos12 x dx tan x u cos12 x dx du
sin x cos x 2sin 4 x
dx
sin 2 x t 2 sin x cos xdx dt
e x dx dt ln t t 2 1
ln e x e 2x 1
C
dt 2 2t 2
1 2
dt 2t 2
1 2
arcsin
1183
1177 dx sin ax cos ax
1 2 t 2
1 4x 2
1181
ex t
t 2 1
dx
2
t
1182
dx
dt
x 2
1
x 2
e u du e u e tan x C
arctan x C
1176 ex
1 2
tdt
1 2
1 T 2
arccos
dt t1t
dt du
ln
dx x4ln 2 x
e dx dt
1 a
dt dx
x
1cos t sin t
1180
ex 1 t
ln
ln x t
dx e x 1
1 a
0 x
sin xdx
T 2
dt t
1174
dx
6xdx dt 5 3
dt sin t
1179
4 3x 2 t
1 a
sin 2t 0 dt sin 2t 0 dt T T
1173
sin t
1178
2 sin x cos xdx dt t
dt 2a
2
2ax t 2adx dt
dx sin 2ax 2
2
dx sin 2ax
dx sin 2 x cos 2 x
2x u 2dx du
dx sin 2 2x 4
4
dx sin 2 2x
2 2
t
1 2
arcsin
2 2
sin 2 x C
du 2
4
2
sin 2 u
du sin 2 u
2 cot u 2 cot 2x C
ln x x 2 1
1184
1
arcsin xx 1x 2
dx
arcsin x 1x 2
arcsin x u dx
udu
2
1x t
du
1x 2
1 2
1x 2
1 2
1 2
u2 t
sec 2 x1
sin x 1 cos x cos x
dx
1 cos 2 x
1
sin x cos 2 x
dx
1cos 2 x cos 2 x
sin x cos x
dx
1cos 2 x
dx
sin x cos x 1cos 2 x
dx
sin x tdt cos xt sin x cos x
dt cos 2 x
C
1 3
cosh tdt
sinh t
1 3
sinhx 3 3 C
dt
arctanh t arctanh 1 cos 2 x C
t 2 1
3 tanh x cosh 2 x
dx
tanh x t
cos 2x 4cos 2 2x
dx
dx cosh 2 x
2x t 1 2
cos t 4cos 2 t
1 2
cos t 51cos 2 t
1 2
du 5u 2 5 5
arctanh
5 10
5 5
u
5 10
5 5
arctanh
sin t
1 2
t 10 dt
:cos 2 x dx sin 2 xcos 2 xcos 2 x :cos 2 x
dx cos 2 x
tan 2 x2
1 cos 2 xtan 2 x2
arctan
t 2 2
x 2 1
2x 5 11 C
x2 2 t
dx
xdx
2 2
arctan
tan x 2 2
C
ln x x 2 1
1 22
dx
1188
x x 2 2
x 2 2
dt 2 2
1 11 t 22
1191a
tan x t dt t 2 2
3 tanh x C
2dx dt arctanh
dx cos 2 x
1 ln 3
2x 5 t
sin 2x C
3t
x2x 5 10 dx
1187* dx 1cos 2 x
1 ln 3
1192 cos t 5sin 2 t
sin t u 1 2
dt
3 t dt
cos tdt du
1 3
1190
2dx dt
3
ln 2 x x 2 1
x3 3 t
1186
2 3
x 2 chx 3 3dx
2 sin x cos xdx 2tdt
5 10
t
dx dt
3x 2 dx dt
1 cos 2 x t 2
2 3
3 2
dx dt
1189
sec x tan x
x 2 1
t dt
arcsin 2 x 1 x 2 C
1 cos 2 x t| 2
x 2 1 x
1
1185
1
x 2 1
dx dt
2 x 2 1
x x 2 1
2xdx dt
dt t
2x
1
x x 2 1
x
t
dt
dt
x 2 2 x
x x 2 2 2 arctan 2 dx
x x 2 2
dx
1 x dx x2
t dt
2 2
dt x2
dt t 2 2
x2 2 C
2 2
arctan
2 2
t
1 t
x 2 dt 2
x 1t 2
dt
2t 2 t2
dt 2t 2
arcsin
2 2
t arcsin
2 2x
C
x ln t dx dtt dtt 1
e ln t 1
dt t
1 t
1
dt t 1t 1
dt 1t
ln1 t ln1
1 ex
C
2
x5x 2 3 dx
10xdx dt
2 3
dt 10x
1 10
t 7 dt
1 8 t 80
1 80
8 5x 2 3 C
x1 t 2 xdt 2 t 1dt
2 3 t 3
2t
2 3
3
x1 2 x1 C
x x 4 x 4 ln1 x C
dx 2x1
dt
2x1
dt
x 2x1
e x dx 2 e x 1
1sin 2 x
t sin x
t 2 1 2
x dt x
dt t 2 1 2
2
dt t 2 1
2 arctanh t 2 arctanh 2x 1 C
dt
2 e x 1 dt ex e x 1
2
2
dt ex
dt t 2 1
2 arctan t 2 arctan e x 1 C
1196
dt cos xdx
arcsinh t arcsinh sin x C
ln 2xdx x ln 4x
ln 2xdx xln 2xln 2
ln 2x t
1192
dx x
x2x 5 10 dx 2x 5 t x
t5 2
t5 10 dt t 2 2
1193 1x 1 x
dx
1 4
t 5t 10 dt
1 12 t 48
5 11 t 44
1 48
2x 5 12
5 44
2x 5 11 C
tdt tln 2 dt tln 2
tln 2ln 2 tln 2
1197
arcsin x 2 1x 2
dt
dx 1x 2
t 2 dt
dx
dt 1 3 t 3
1 3
dt
t ln 2 lnt ln 2 ln 2x ln 2 lnln 2x ln 2 C
arcsin x t
x t dx 2 x
t
txdt xtln 2
dt ln 2
2dx dt
t 2 4t 4 ln1 t
ex 1 t
cos xdx
2 3 t 3
dx e x 1
1191e
1t 2
1195
dt 2
dt t1
2
2x 1 t 2x 1 t 2
xdx x1
dt
3
dx x 2x1
1191d
dt
2 t1
1194
5x 2 3 t
dx 2 x1
tt 3 1t
t3 t2 t 2 t t 2 t 2t 2t 2 2
7
tdt 2
2 t 2 dt tdt 2 dt 2
1191c
x5x 2 3 7
1t 2 1t
t t : t 1 t t 2
dx e x 1
2 x dt 2
1x 1 x
3
1191b
arcsin 3 x C
cos t u
1198
e 2x e x 1
sin tdt du
ex 1 t e x dx 2 e x 1
x e 2x 2 e 1 t ex 3 x
2 3
dt 2
t 2 1t t
dt 2 t 2 1dt
2 3 t 3
x 2 a 2 x
sin 3 x cos x
5
2 5 t 5
2t
1200* dx
z 2 dz z 2 a 2
x 2 a 2 a arctan
2
t|
2
dx
dz a 2
dz z 2 a 2 x 2 a 2 a
z a arctan
z a
sin t a arctan
sin t a
C
tdt x
xt
1 x2
dx dt t2
dt
1 t 2 1
2
dt arctanh t arctanh 1 x C
1201
1 t
arcsin x2
1x 2
dx
cos 2 t 1cos 2 t
x3
sin tdt
arccos x
1 4
cos 2 t sin t sin t
dt cos 2 tdt
1cos 2t 2
dt t
1t 2 t2
1 t
1t 2 t2
dt t 1 t
1t 2
dt 1t 2
arcsin t
C dx
1 2
t
1 4
sin 2t
dx cosh t sinh 2 t1
sin2 arccos x C
sinh 2 t1
t t1 sinh t cosh t sinh t cosh t cosh sinh sinh t sinh t sinh t t arcsinh x cosh t 2 arctanhe cosh arcsinh x 2 arctanhe C
dx
2
2
dt sinh t
1206*
2
2 x cos t
dx x 2 4x 2
2xdx sin tdt x3 cos t sin t cos t
1 x
1
dt t2
x sinh t
1202 2x 2
1 t2
x 2 1 x
dx sin tdt 1 2
dt t2
1205
x cos t
1 t
x
xdx tdt
sin 2 t cos tdt sin 2 ta 2
x x 2 1
1 x2 t2
sin 2 t cos tdt x2
1x
1204*
x 1x 2
sin t sin t cos tdt x x
cos tdt dz
cos x 2 cos x C
dx
sin t z
2 sin 2 xdt 2 1 cos 2 xdt 2 1 t 4 dt
3
2 x 2 2 C
xdx sin t cos tdt
dx
x dx dt 2 sin cos x
1 3
x a 2 sin 2 t
cos x t
2 5
3
3
u du 2 u 13 u 2 2 cos t 13 cos t 2
2
1199
1 2
1203
2t
2 ex 1 C
e 1
2 2 x2
dt
du u
sin tdt 2x
dt
1 2
x 2 sin t x2 cos t
sin tdt 2
cos t sin tdt
2cos t sin tdt 2 cos t
2 sin tcos t sin t 2 cos t
dx 2 cos tdt
dt
2 cos tdt 2 sin t 2 42 sin t 2
2 cos t 8 sin 2 t cos t
dt
1 4
2 cos t 4 sin 2 t 44 sin 2 t dt sin 2 t
1 4
dt
2 cos t 4 sin 2 t 44 sin 2 t
tan t
1 4
dt
tanarcsin
x 2
2 cos t 8 sin 2 t 1sin 2 t
C
dt
2 cos t 8 sin 2 t cos 2 t
dx
1210
x 2 4x 2
x
dx
dt t2
1 t2
1 4
4
1 t dt t2 dt 4
1 t2
dt 1 t
4t 2 1
1 2
t t 2 14
dt
1 2
1 2
2
4t 1
4t 2 1 2
1 4
4 1x 1
1 4x
dx a sinh tdt
1x 2
dt a x a
a cosharccosh
1x 2
dx
a 2 cosh 2 t
a cosh 2 t1 dt a sinh t
dt a
cosh 2 t cosh 2 t1
dt
sinh tdt a cosh t a ln tanh
a ln tanh
arccosh 2
x a
x a lntanh
1 2
t 2
arccosh
x a
C
ln xdx
ax b 1 x
1x 2
ax b 1 x 2
1x 2 1x 2
a 1 x2
1x 2
c
2
1x 2 1x 2
2xaxb 2 1x 2
1x 2
|
dx dv ln x u
1x 2 c
1x 2
1 2
I
x 1 x2
1 2
dx 1x 2
1 2
x ln x x
| 1 x2
du x ln x x C
arctan xdx arctan x u dx dv dx 1x 2
x 1 x2
1 2
du
x arctan x
arcsin x C
xv x 1x 2
dx
1 x2 t 2xdx dt
dx x1x
x arctan x 2
x sin t 2 sin t cos t sin 2 t1sin 2 t
dt 2
sin t cos t sin t 1sin 2 t
dt 2
sin t cos t sin t cos 2 t
dt 2
sin t cos t sin t cos t
dt 2 dt 2t
x arctan x
t
1 2
ln t x arctan x
1 2
1 2
ln1 x 2 C
arcsin xdx arcsin x u dx dv
2 arcsin x C
dx 1x 2
1209
du
x arcsin x
a 2 x 2 dx
xv x 1x 2
dx
1 x2 t
x a sinh t
2xdx dt
dx a cosh t
x arcsin x
a 2 a 2 sinh 2 t a cosh tdt a 2 1 sinh 2 t cosh tdt a 2 cosh 2 t cosh tdt a 2 cosh 2 tdt a 2 cosh22t1 dt a2 cosh 2tdt 14 a 2 sinh 2t 12 a 2 t 14 a 2 sinh2 arcsinh ax 12 a 2 arcsinh ax 2
x x 2 a 2
dt 2
1213
dx 2 sin t cos tdt
1 2
dx x
1212
1208
dx x
xv
c
1 x 2 a1 x 2 xax b c 1 x 2 a 2ax 2 xb c 1 x 2 x 2 2a xb a c a 12 , b 0 a c 1. . . . c 12
a 2 cosh 2 ta 2 1sinh 2 t dt sinh t
1211
1 x 2 dx
a 2 cosh 2 t
a
4 x2 C
1207
dx
x a cosh t
1 t2
x2 x 2 a 2
1 2
a 2 arcsinh
x a
C
a2 2
dt
1214
x sin xdx
dt 2
t
x arcsin x
dt t
x arcsin x t x arcsin x 1 x 2 C
xu
sin xdx dv
tu
e t dt dv
dx du
cos x v
dt du
et v
x cos x cos xdx x cos x cos xdx x cos x sin x C
te t e t dt te t e t 1 2 t 2 2 t I 271 t 2 e t 2I 1 271 t 2 e t 2te t e t 27 t e 27 te t 27 e 2 3x 1 2 2 3x 1 2 3x 2 2 3x 3x 3x 27 3x e 27 3xe 27 e 3 x e 9 xe 27 e C
1215
x cos 3xdx
1219
3x t
x 2 2x 5e x dx x 2 e x dx 2 xe x dx 5 e x dx
3dx dt
1 9
t cos tdt tu
cos tdt dv
dt du
sin t v
19 t sin t sin tdt 19 t sin t 19 cos t 19 3x sin3x 13 x sin 3x 19 cos 3x C x ex
1 9
cos3x
xe
x
e x v
e dx xe x e x C
e x dx dv
dx du
e x v
x
x3e 3
x2 x dx
3x t dx3 dt
2 x dx dv
xu
27t 3 e t 3dt 81 t 3 e t dt 81I 1
dx du ln12 2 x v ln12 x2 x ln12 2 x dx ln12 x2 x 1218
x 2 e 3x dx
et
dt 3
1 27
t 2 e t dt t2 u
et v
2tdt du
t 2 e t 2 e t tdt
I 1 e t tdt
1 ln 2
x2 x
1 ln 2 2
2 x C
I 1 t 3 e t dt e t dt dv
t3 u
et v
3t 2 dt du t 2
I 2 e t t 2 dt
e t dt dv 1 27
2 x dx
t e 3 e t dt t 3 e t 3I 2
3dx dt t2 9
1 ln 2
3 t
3x t
I
xu
1220*
x
1217
e x v
I 12 e x e x dx e x x e x dx e x x e x I 1 e x x 2 2e x x e x e x x 2 2e x x 2e x I 2 2 xe x dx 2e x x e x 2e x x 2e x I 3 5e x I I 1 I 2 I 3 e x x 2 2e x x 2e x 2e x x 2e x 5e x I e x x 2 5e x C
e x dx dv
dx du
2xdx du
x
dx xu
e x dx dv
I 1 e x x 2 e x 2xdx e x x 2 2 xe x dx
1216
x2 u
e t dt dv
t2 u
et v
2tdt du
I 2 t e 2 e t tdt t 2 e t 2I 3 2 t
I 3 e t tdt e t dt dv
tu
et v
dt du
te t e t dt te t e t
I 2 t 2 e t 2I 3 t 2 e t 2te t e t t 2 e t 2te t 2e t I 1 t 3 e t 3I 2 t 3 e t 3t 2 e t 2te t 2e t t 3 e t 3t 2 e t 6te t 6e t I 81I 1 81t 3 e t 3t 2 e t 6te t 6e t x x x x 3 2 81 3x e 3 243 3x e 3 486 3x e 3 486e 3 3x 3 e
1 3
27x 2 e
x
1 3
x
162xe
1 3
x
486e
1 3
x
ln x u x 2 dx dv x3 3
1 2
x sin 2xdx
ln 2 x u 2 ln x x
2x t
v
x3 3
ln x
1 3
x 2 dx
1 3
x 3 ln x
1 9
x3 C
1 2
x ln x x
sin t
dt 2
ut
dv sin tdt
t 2
1 8
dx dv
dx du
2
2dx dt
xv
2 ln x x
dx x ln 2 x 2 ln xdx
I 1 ln xdx
t sin tdt
ln x u dx dv dx x
du dt v cos t t cos t cos tdt t cos t cos tdt t cos t 18 2x cos2x 18 sin2x 14 x cos 2x 18 sin 2x C 1 8
1 8
1 8
1 8
sin t
du
xv
I 1 x ln x x dxx x ln x x I x ln 2 x 2x ln x x x ln 2 x 2x ln x 2x C 1225
1222*
x 2 5x 6 cos 2xdx
ln x x3
dx x
2dx dt t2 4
3 sin t
5 t 2 1 8
6 cos t
dt 2
t 2 cos tdt
5 4
1 8
t cos tdt 2
t cos tdt
5 4
t cos tdt 3 cos tdt
ln x
cos tdt dv
tu
cos tdt dv
2tdt du
sin t v
dt du
sin t v
ln x x
I b t sin t sin tdt t sin t cos t I aa t sin tdt sin tdt dv
dt du cos t v t cos t cos tdt t cos t sin t I a t 2 sin t 2t cos t sin t t 2 sin t 2t cos t 2 sin t I 3 sin t 18 I a 54 I b 3 sin t 18 t 2 sin t 2t cos t 2 sin t 54 t sin t cos t sin t 18 t 2 sin t 14 t cos t 54 t sin t 54 cos t 114 sin2x 18 2x 2 sin2x 14 2x cos2x 54 2x sin2x 54 cos2x 114 sin 2x 12 x 2 sin 2x 12 x cos 2x 52 x sin 2x 54 cos 2x C
x 2 ln xdx
x 2 2
v
x 2 dx 2 x
12
ln x x2
1 2
dx x3
1 ln x 2 x2
1 4x 2
C
dx
ln x u
I a t 2 sin t 2 t sin tdt
1223
x 2 2
du
1226
t2 u
tu
dx
ln x u x 3 dx dv
2x t
11 4
ln x
x 3 dx 3 x
ln 2 xdx
x sin x cos xdx
x3 3
1224
C
1221
du
dx x
dx x
dx x
dv
du 2 x v
2 x ln x 2 x
dx x
2 x ln x 2
dx x
2 x ln x 4 x C
1227
x arctan xdx arctan x u xdx dv dx 1x 2
x2 2 x2 2
x2 2
du
arctan x arctan x
1228
x arcsin xdx
1 2
v
x 2 dx 2 1x 2
dx
x2 2
1 2
arctan x
dx 1x 2
1 2
1 2
x2 1x 2
dx
x 2 arctan x
1 2
x2 2
x
arctan x 1 2
1 2
arctan x C
x 2 11 1x 2
dx
e x sin xdx
xdx dv arcsin x u 1 2
dx
x2 v
du
1x 2
1 2
x 2 arcsin x
1 2
x 2 arcsin x
1 2
1 2
1 2 1 4
2
x arcsin x
dx
x2
1x 2 1x 2 1
dx
1x 2
2
x 1 x
1 2
x 2 arcsin x
1 4
arcsin x C
1 2
1 x 2 dx
1 2
dx 1x 2
dx 1x 2
I 1 e x cos xdx
x ln x 1 x
dx 1x 2 x
2
1x 2
x ln x 1 x 2
x 1x 2
dx
dx
2xdx dt
2
x ln x 1 x 2
ex u
sin xdx dv
e x dx du
cos x v
e x cos x cos xe x dx
t
t x ln x 1 x 2
1 x2 C
ex u
cos xdx dv
e x dx du
sin x v
I e cos x e x sin x sin xe x dx I e x cos x e x sin x I 2I e x cos x e x sin x x x sin x I e cos xe C 2 x
xdx sin 2 x dx sin 2 x
xu
dv
dx du cot x v x cot x cot xdx
1233
x cot x cot xdx x cot x lnsin x C
3 x cos xdx
1231
3 x dx dv
I
e x sin x e x cos x I
e cos x cos xe x dx
dt 2
1230
x cos x sin 2 x
cos x sin 2 x
3x ln 3
dx dv
sin1 x v I1
xu
x
3 dx dv
cos x sin 2 x
I x
1 sin x
sin x du
dx du
3x ln 3
sin x t dt t2
v
1 t
I
1 sin x
I
1 sin x
dx
cos x u
x x ln3 3 cos x sin x ln3 3 dx x I ln3 3 cos x ln13 3 x sin xdx
cos xdx dt
1232
sin xdx du
x
x ln x 1 x
cos x u
ex v
I e x sin x e x cos x e x sin xdx 2I e x sin x e x cos x x x cos x I e sin xe C 2 x e sin xdx
1 x2 t
e x dx dv
I 1 e x cos x e x sin xdx
u
du
x
x ln x 1 x 2
cos xdx du
e x cos x e x sin xdx
dx dv ln x 1 x 2 xv
sin x u
ex v
I e x sin x e x cos xdx
1229
ln x 1 x 2 dx
e x dx dv
x sin x
dx sin x
x sin x
ln tan
x 2
C
3x ln 3 3x ln 3
I 1
v cos x cos x
1 ln 2 3
sin x u cos x du
x 1 sin x ln3 3 ln 3 3x sin x ln12 3 ln 2 3 x x 3 cos x ln32 3 ln 3
3x ln 3
I sin x
cos x
3x ln 3
cos x
3x ln 2 3
sin x
1 ln 2 3
3 x cos x
3x ln 3
I
3x ln 2 3
cos x
sin x
ln 2 31 ln 2 3
3x ln 3
cos x
3x ln 2 3
sin x
ln 2 3 3 x
ln 2 31
cos x ln 3sin x ln 2 31
C
1234
1236
x 3 e x 2 dx x 2 t
e ax sin bxdx
2xdx dt
bx t
x3et
bdx dt at
e b sin t dtb e
e cos t e a b
I I I 1 I I
sin tdt dv
at b
1 b
I
u
t
e a b dt du cos t v
a b
at b
e
at b
a bt
1b 1b 1b 2 ab 2 1 b
e
t b
e a cos tdt
a b
u
e cos t at b
e cos t at b
e cos t
at b
a b2 a b2 a b2
e
b cos bxa sin bx a 2 b 2
sin t
1 b
e
a bt
at b
e sin t a b2 t
2
e x C
dx 2tdt
e sin t aI
ea b
1 2
x t2
sin tdt
a2 b2
2 e t tdt
I
at b
e sin t
b cos ta sin t a 2 b 2
ea
bx b
b cosbxa sinbx a 2 b 2
e ax C
tu
e t dt dv
dt du
et v
2 te t e t dt
2te t 2e t 2 x e
x
2e
x
C
1238
x 2 2x 3 ln xdx
sinln xdx
x 2 2x 3dx dv ln x u
ln x t
x3 3
dt
x3 3 x3 3
x sin tdt e t sin tdt
e t dt dv
sin t u
et v
cos tdt du
x ln
I e sin t e t cos tdt
et v
sin tdt du
I 1 e t cos t e t sin tdt e t cos t e t sin tdt I e sin t e cos t e sin tdt e sin t e cos t I 2I e t sin t e t cos t t t cos t ln x ln x cos ln x I e sin te e sin ln xe 12 x sinln x 12 x cosln x C 2 2 t
t
du
x3 3
x 2 3x dxx
x 3x ln x x 2 3x ln x
1x 1x
x3
9
x2 2
x3 3
x 2 3x ln x
3x C
t
t
x2 2 x2 2
ln ln
1240
ln 2 x x2
dx
1x 1x
2 x 2 1
cos t u
dx x
2
ln
I 1 e t cos tdt e t dt dv
x 2 3x v
1239
t
t
te t e t dt 12 te t 12 e t 2 2 2 x 2 e x 12 e x 12 x 2 e x
1 2 1 2
at b
1235
dx x
et v
x t| 2
e sin t a
1b e cos t at b
dt du
te t dt
1 2
e x dx
at b
at b
cos t a2 b a 2 b 2 b2
e t dt dv
sin t v
at b
x 2 e t dt
1237
cos tdt dv
dt du
1 2
tu
at
e b sin tdt
1 b
dt 2x
dx
u
dv xdx v
x2 2
x2 2 2 x 2 1
x2 2
dx x 2 1
dx du 1x 1x 1x 1x
dx
2 1x x 2x1 1x 1 2 x ln 1x x 2 1x
ln
x2 2
ln
1x 1x
arctanh x C
x 2 11 x 2 1
x2 3
x 3 dx
ln 2 x u 2 ln x x
1 x 1 x
dx x2 1x
dx du v
ln x 2
arctan 2 x u
dv
1x 2 lnx x 2 lnx 2x dx
2 arctan x 1x 2
dx
ln x 2
ln x u dx x
du
dx x2
1 x
ln 2 x
2 x
ln x
2 x
C
x2 2 x2 2 x2 2
dx
lnln x u dx x ln x
du
dv
ln x v dx x
ln x lnln x ln x C
dx dv
dx x arctan x v
arctan 2 x x arctan x arctan 2 x 2
x 1x 2
dx
arctan x 1x 2
dx
2
1 2
ln1 x 2
arctan x t dx 1x 2
3dx dt
t2 9
arctan t dt3
t 2 arctan tdt
1 27
1 27 1 27
t3 3 t3 3
t3
du arctan t
dt
1 81
t 3 arctan t
t3 1t 2
vx
dt
2tdt dz
Iv x
3 1 t 3 arctan t tz dz 81 2t 2 1 1 t 3 arctan t 2 tz dz 811 t 3 arctan t 12 z1 z dz 81 dz 1 1 1 1 3 3 t arctan t 2 dz 2 z 81 t arctan t 12 z 12 81 1 t 3 arctan t 12 1 t 2 12 ln1 t 2 81 1 3x 3 arctan3x 12 1 3x 2 12 ln 1 3x 2 81 1 27x 3 arctan 3x 12 92 x 2 12 ln1 9x 2 81 1 3 1 1 x arctan 3x 162 181 x 2 162 ln1 9x 2 C 3
x arctan 2 xdx
2
arcsin x 1x 2
I x arcsin 2 x x2
1 t2 z
1243
arcsin 2 x u
dv dx
arctan t
t 3 dt 3 1t 2 1 t3 3 1t 2
12 arctan 2 x x2 C
I arcsin x 2 dx
v
3
1 2 t 12 arctan 2 x 2 x2 2 arctan x x arctan x arctan 2 x 12 ln1 x 2 2 1 2 x arctan 2 x x arctan x 12 arctan 2 x 12 ln1 2
1244
arctan t u t 2 dt dv dt 1t 2
dt
tdt
3x t
x 2 arctan x 1x 2
dx arctan 2 x x arctan x arctan x x arctan x 1x 2
dtt 12 ln t x I 2 arctan dx 1x 2
x arctan 3xdx
arctan 2 x
2xdx dt
2
du
1 2
1242
x2 1x 2 x 2 11 1x 2
x2 2
1 x2 t dx x
ln x lnln x ln x xdx ln x lnln x ln x
v
x 2 2 arctan x 2 1x 2
arctan x x arctan x arctan x I 1 I 2 I 1 1xx 2 dx
1241 lnln x x
arctan x 2
dx 1x 2
1x ln 2 x 2 1x ln x
x2 2
arctan x u
dv dx x2 v 1x
1x ln 2 x 2 1x ln x 1x dxx
x2 2
xdx dv
dx du
ln z
arcsin x 1x 2
dx
arcsin x 1x 2
arcsin x u
dx du
1 1x 2
x 1x 2
dx dv
dx du 1 x 2 v
I v uv vdu 1 x 2 arcsin x 1 x 2 Iv
x 1x 2
1 x2 t 2xdx dt
1 1x 2
dx 1 x 2 arcsin x x
1 2
1
1
12 t 2 dt 12 2t 2 t 1 x 2
dt t
1247
I x arcsin 2 x 2 1 x 2 arcsin x x x arcsin 2 x 2 1 x 2 arcsin x 2x C
x tan 2 2xdx
1245
2x t
arcsin x x2
dx
2dx dt dx x2 1x
arcsin x u dx
du
1x 2
I1
tdt x
xt
1x arcsin x
dx
x
I 1x arcsin x I 1
dt x2
1 x
dt 1t 2
arcsin x 1x
dx x 1x 2
arctanh t arctanh 1 x 2
sin 2 x ex
dx
sin 2x u tu
I 2 t 12 arcsin 2 t
1 2
I 1 arcsin tdt
arcsin 2 t u
cos 2x u
t arcsin 2 t arcsin 2 tdt
arcsin 2 tdt
arcsin t u dt 1t 2
1t 2
dt
tdt 1t 2
dv
1249
dv
cos 2 ln xdx
du v 1 t 2
I 2 1 t arcsin t 1 t 2
dx ex
I 2 e x cos 2x 2 e x sin 2xdx e x cos 2x 2I 1 I 1 e x sin 2x 2e x cos 2x 2I 1 x x cos 2x e x sin 2x 2e x cos 2x 4I 1 e sin 2x2e 5 x x cos 2x I e x sin 2 x I 1 e x sin 2 x e sin 2x2e C 5
tv
t arcsin t
v
2 sin 2xdx du e x v
dt dv
dt du
dv x
e x sin 2x 2 e x cos 2xdx I 1 e x sin 2x 2I 2
2
t arcsin 2 t 2
dx ex
2 cos 2xdx du e
dt dv
arcsin 2 t v dt du
1t 2
sin 2 x u
dv
I 1 e x sin 2xdx
dt
1t 2
2 arcsin t
dx
I e x sin 2 x e x sin 2xdx e x sin 2 x I 1
t arcsin t
1t 2
ttan t t tan t tdt 14 t tan t t 2 tan tdt tdt t tan t 12 t 2 lncos t 14 2x tan 2x 2x 2 lncos 2x C
e x v 2 sin x cos xdx sin 2xdx du
dx 2tdt
arcsin t
1 4 1 4
dx ex
x t2
1 2
I I 1248
arcsin x arctanh 1 x 2 C
x t
I 2
ut
tan t t v du dt
1246
t tan 2 tdt tan 2 t dv
xdx tdt
1x 2
dt x
1 4
1 x2 t2
v
1x arcsin x 1x
1 x2 t
dv
2
dt 1t 2
arcsin t 1 t dt 2
dx dv
2 cos ln x sin ln x dxx du
xv
I x cos 2 ln x x2 cos ln x sin ln x dxx x cos 2 ln x x2 cos ln x sin ln x dxx
2
I 2 arcsin t 1 t t I 1 t arcsin 2 t 2 arcsin t 1 t 2 t
cos 2 ln x u
t arcsin 2 t 2 arcsin t 1 t 2 2t
I x cos 2 ln x sin2 ln xdx x cos 2 ln x I 1 sin2 ln x u
I t arcsin 2 t I 1 t arcsin 2 t t arcsin 2 t 2 arcsin t 1 t 2 2t 2
2
I 2 arcsin t 1 t 2t 2 arcsin x 1 x 2 x I 2 1 x arcsin x 2 x C
2 x
cos2 ln xdx du
dx dv xv
I 1 sin2 ln xdx x sin2 ln x x 2x cos2 ln xdx
I 1 x sin2 ln x 2 cos2 ln xdx x sin2 ln x 2I 2
a 2 a 2 sin 2 t a cos tdt
I 2 cos2 ln xdx
a 2 1 sin 2 t cos tdt a 2 cos 2 tdt a2 1 cos 2tdt
I2 I2 I1 I1
2
cos2 ln x u
dv dx
2x sin2 ln xdx du
vx
a2 2
x cos2 ln x x sin2 ln xdx x cos2 ln x 2 sin2 ln xdx x cos2 ln x 2I 1 x sin2 ln x 2x cos2 ln x 2I 1 x sin2 ln x 2x cos2 ln x 4I 1 x sin2 ln x2x cos2 ln x 5 2 x
x sin2 ln x2x cos2 ln x 5
I x cos 2 ln x
x
dv
1 1x 2
1 2
I
dx 1 2 a2 x 2 a 2 a13 arctan ax
x 2 a 2 x 2 1 2 dx a 2 x 2 a 2 x2 a12 2 dx x 2 a 2
dx x 2 a 2
1 a2
x2 2 x 2 a 2
dx
I I
1 a3
arctan arctan
x a
t 2 t 2 1
1 a2
1 a3
dt dv
a2t2 2 a 2 t 2 a 2 t2 2 dt t 2 1
adt
I
1 a3
arctan
x a
1 a3
I
1 a3
arctan
x a
1 2a 3
1tt 2 arctan t
I
1 a3
arctan
x a
1 2a 3
dx a cos tdt
a
t2 2 a 4 t 2 1
1 21t 2 t
ax
x A
sinh 2 arcsin
A 2
arcsin
x A
cosh 2t 1
C
Ax 2 Ax 2
2xaxb
a A x2
1 2
dx Ax 2 c
2 Ax 2
A 2
,c
I 12 x A x 2 dt
1 2
I
v dt du
x a sin t
arctan
x a
1 ax 2
Ax 2 Ax 2
A 4
t
A 2
Ax 2
|
| A x2
A 2
x A x2
1 2
A 2
dx Ax 2
A ln x A x 2
C
1254*
1 21t 2
a 2 x 2 dx
1 a3
tu
1252**
A 2
ax b A x 2 c
Ax 2
b 0, a
dx adt x a
sinh 2t
c A aA A1 a
x at 1 a3
C
A x 2 aA x 2 xax b c A x 2 aA 2ax 2 xb c aA c A 2a 1
arctan x C
1251*
A 4
Ax 2
x dx 22x 2
x a
A cosh tdt
A x2
1 1 21x 21x dx 2 x 2 1 2
a 2 sin2 arcsin
A A sinh 2 t A cosh tdt A 1 sinh 2 t cosh tdt A cosh 2 tdt
1 dx du 21x 2 v
x 21x 2
1 4
2
A sinh t
dx
C
dx xdx 2 x 2 1
1 4
2
A x2
xu
cos 2tdt a t a sin 2t 1 2
1253*
1250** x2 2 x 2 1
t a 2 arcsin
1 2
a2 2 x a
1 21t 2
dt
1 a3
arctan ax C
arctan
x a
1 2a 3
1tt 2
dt 1t 2
x2 9x 2 9x 2 9x 2 9x 2 9x 2
9 x2 9 x2 9 x2 c 9a a I1
x 2
dx
9x 2 9 9x 2
dx 9 x 2 9
ax b 9 x 2 a 9 x2
axbx
9x 2
c 9x 2 c 9x 2
dx 9x 2
9 x 2 9 arcsin
dx| | 9 x2
a9 x 2 ax bx c 9a 2ax 2 xb c x 2 2a xb c 9a 9. . . c 92 9
1 2
, b0 , c
9 x2
I 9 arcsin
x 3
9 2
dx 9x 2
9 2
I 1 9 arcsin
x 3
x 2
9 x2 x 2
9 2
arcsin
9 x2
9 2
1 3
x
arcsin
x 3
x 3
9 2
I
x 3
arcsin
x 2
1255
dx x 2 2x14
dx x1 2 4
1 2
arctan
x1 2
C
x2 x 2 6x10
dx x 2 2x11
dx x1 2 1
arctanhx 1 C
1 3
xdx x 2 7x13 7 2
x
dx x 2 13 x 13
1 3
dx x 16
2
1 3
1 1 36 3
dx 2
x 16
11 36
2 11
arctan
2
x 72
xdx x 72
49 13 4
x 72
2
dx
34
7 2
11
x 3
C
t t 2 34
34
t
dx 2
7 2
2
x 72
1 2
34
ln4t 2 3
7 3
1 2
ln 4x
1 2
lnx 2 7x 13
3
arctan 2x7 3 7 arctan 2x7 3 3
7 3
1 3
3 arctan
t 1 z
dx dt
2tdt dz
t t 2 1
dt 4
dt t 2 1
3
dx
x
5 2
x
5 2
2x 65
x 2 2x1 x 2 3x4
x 2 3x4 2x3 x 2 3x4
x
t
9 2
2
x 3x 4 t x 2x 3dx dt x
5 2
dt t
9 2
5 2
3
2
dx
xdx x2 2 1 dx 6 x2 2 1
3 2
dt 4 arctanh t
x 2 3x42x13x4 x 2 3x4 2x 65 33 x 32
dz z
dx x2 2 1 dx 2 x2 2 1
3 2
dt 33 16
1 2
dx 3 2
32 xx 2
1 2
4t 33
dx 33 16
x 34
2
1 2
t 2
arcsin
1 2
arcsin
4x3 33
C
dx 1 4
x 12
2
t
dx dt
dt 1 4
arcsin 2t arcsin2x 1 C
t 2
2
94 4
dx
x
5 2
2
p 2
x 2 3x4 x 2 3x4 2x3 95 x 32
2
dx
74
5x3 x 2 3x4
ln
1 2
7 arctan
p2 4
q
dx p
x 2
2
p2 4
dx p
q
x 2
2
p 2 4q 4
t
dt t 2 z
z ln t t 2 z 2
p x x xp q 3x6 x 2 4x5
dx
3 2
x 2 4x 5 t 9 7
2 7
z 7
ln x
C
1265
ln t
2
dx dt
z 5 2
dx p
x 2
p 2 4q 4
74
x
dx x 2 pxq
x
dx dz dz z 2 74
x 3 ln t 8 arctan z
t
dx xx 2
x
4 arctanh t
dx x 32
x3 2 1
1264
t 2 1
dx
2x6 10 6 3
dx dz
dz z 2 1
1 2
3 4
1 2
1260 x1 2 x 2 3x4
8
dt t
dx
32 ln z 4 arctanh t 32 lnt 2 1 4 arctanh t ln x 2 2 1 4 arctanhx 2 32 lnx 1 32 lnx 3 4 arctanhx 2
dx x 3
1263
C
2
x2 t
6x10 x3 2 910
dx dt
2x 7 3
3x2 3x2 3x2 dx dx dx x 2 4x3 x2 2 43 x2 2 1 dx x22 x2 3 dx 2 3 x2 2 1 x2 2 1 x2 2 1 dx x2 3 dx 4 x2 2 1 x2 2 1
3 3 2
33x2x 2
x 7 2
dt
dx
1262 2
1259
C
x 3 lnx 2 6x 10 8 arctanx 3 C
dx x 72
6x1
dx dt
ln 2
x 2 6x106x10 dx x 2 6x10 2x6 dx dx 8 x 2 6x10 x3 2 1
2x 6dx dt
dx 3x 2 x1
2x3 7
x 2 6x 10 t x 3 z
1258
arctan
dx
x 3
dx x 2 2x
1257
9 7
lnx 2 3x 4
1261
dx x 2 2x5
1256
5 2
x
9 x2 C
2x 4dx dt
2x4 x 2 4x5
dx
p 2
x
p 2
2
p 2 4q 4
dx
3 2
dt t
dx
2x8 1xx 2
5 4
5 4
x
1 2
x 1 2
2x
14
x
2
dx dt dz 5 4
x4 5 4
1 2
1 2
dx 2
5 4
x 12
2
dx
z 2 5
5 z 2 1 x x 2 9 arcsin
2x1 5
x 5x 2 2x1
1 10
dx
10x22
1 10
dt t
2x x 5
C
1 5
1
1 10
2t 2
1 5 5
10x2 5x 2 2x1
dx
1 5 5
dx x 15
2
1 1 25 5
x dz 4 z 2 25
2
5 25
arcsinh
5 2
z
5x 2 2x1 5
5 25
arcsinh
5x1 2
dt 1t 2
2
arctanh t arctanh 1 x C
arcsin
1x1 x1
2
arcsin
arcsin
2x x1
2
C dx
x1 x1 2 1
1 z
dz z2 1 z
2
1z 1 1 t
dz z 2
1z 1
arcsin
1 x1
dz z 1z 2 z2
dz 1z 2
arcsin z
C
1272
dx
x 2 2x 5 dx x 2 2x 1 4 dx x 1 2 4 dx 1 t
x
x1 t
dx dt t2
dt t2
1 t2
1t 1 1 2
dx dt
dtt 1 t2
1t 1
dtt 1tt 2 t2
dt t 1 t
1tt 2
z
dz 5 4
z 2
dt 1tt 2
dt 1 t 12
2
14
t 2 4 dt t 2 sinh z dt 2 cosh zdz
dt dz
arcsin
x x 2 x1
t
1
2
dz 2z1 2
dt
1269
1 t
1 x1
dt dz z2
xdx tdt
arcsin
2 2x 2 x1
dz 12zz 2
t t 2 1
t
1 x2 t2
xt
1
12zz 2 z2
dx dt
tdt x2t
1 t
dzz
x1 t
1 x2 t
2z 1
dx
1x 2
tdt x
dz z2
1 z2
x1 x 2 2x
C
dx x
1271
4 25
t 5
1 z
arcsin
dx 1 5
1 z
arcsin
z
dt t t 2 2t1
dt dz z2
C
dx dz
1 5 5
dx
x 2 25 x 15
1 10
dt t t1 2 2
t
1268
2x arcsin 2x
dx dt
10x 2dx dt
2 5
dx
dx dz
5x 2 2x 1 t x
arcsin
x1 t
1267
1 2
x1 x 2 2
2
2 t 9 arcsin
z 2
2
x 12
x 12 4 12
dx 5 4
t x
9
dt t
2
dx 9
2
1 2
dx 2
2x8 1 x 12
x 12
2
1x
1270
1266
2 5
arcsin
3 t 3 x 2 4x 5 C
4 sinh 2 z 4 2 cosh zdz 4 sinh 2 z 1 cosh zdz 4 cosh 2 zdz arcsin
2 5
z arcsin
2 5
t
1 2
4
1 2
1 cosh 2zdz 2 1 cosh 2zdz 2 dz 2 cosh 2zdz
z1 2
2z u
1276
2dz du
2z cosh udu 2z sinh u 2z sinh 2z 2 arcsinh 2t sinh2 arcsinh 2t 2 arcsinh
x1 2
sinh2 arcsinh
1273
x1 2
x
1 2
1 4
dx
1 4
1 4 1 4
t 2 t 2
t 2
1 4
1 4
t 2 dt
1 4
1 4
1 8
t2 1 2
t2
1 4
2tatb 1 4
2
t 2
c 1 4
c
1 4
t 2
1 4
|
dt 1 4
t 2
1 4
t 1 4t 2
1 8
arcsin 2t
1 8
arcsin 2x
1 2
2
1 4
2x 1 x x 2
1 8
arcsin2x 1 C
2 x x 2 dx 2
1 4
1 2
1 4
2
x 2x x
9 8
t 9 4t 2
9 4x
9 4
x
1 2
1 4
1 2
2
9 8
arcsin arcsin
2
2x x
9 8
2 3 2 3
t x
arcsin
1 2
dt t 2 1
2
14
sin x cos x2 2 3
dt 3 4
t 12
2
arcsinh
2t1 3
arcsinh
xdx x 4 4x 2 41
ln xdx
ln cos x 2 cos 2 x 4 cos x 1
ln xdx x ln x2 2 41
ln xdx x 5ln x2 2
2
du
I
udu 5u2 2
u2 z
2 3
x
z2 5z 2
1 3
C
arctanh t
1 2
arctanhx 2 2 C
zdz 5z 2
2
5 z2 t
zdz
2zdz dt
5z 2
dz 5z 2
dt 2
t
t 5 z2
I 1 5 u 2 2 1 u 2 4u 1 ln 2 x 4 ln x I2
xdx 2 x 2 2 1
dz
dz 5z 2
arcsin
z 5
arcsin
u2 5
arcsin
I I 1 2I 2 1 ln 2 x 4 ln x 2 arcsin 1280
1 2
2e x 1 3
C
dx
ln t t 2 3
x 14 ln xln 2 x
I1
2xdx dt 1 2
x2 2 t
dt 1 t 12
du dz
t 2 dt
xdx x 4 4x 2 3
dt t 2 3
dx x
1275
sin xdx cos 2 x4 cos x1
t
1 2
ln x u
x x 2 dx
1 4
dt
I
dx dt
x
C
1279
1274
1 4
sin x3 3
dx
1tt 2
1 2
arctan
cos x 2 t
1 4x
9 4
1 3
sin xdx dt
x
1278
t2
1 4
1 2
ex
dt|
t 2
x
t 3
e dx dt
t 2 a 14 t 2 tat b c t 2 t 2 2a tb c 14 a a 12 , b 0 . c 14 a 14 . . . . c 18 14 c 18 t 2
arctan
x
1 4 1 4
I
1 3
1e x e 2x
dt
t 2
t2
dt t 2 3
ex t
t 2
dt at b a
t 2
1 4
dx
1277
t
dx dt 1 4
cos x sin x3 2 3
cos xdx dt
2
1 2
x
dx
sin x 3 t
C
x x 2 dx
cos x sin 2 x6 sin x12
I
dx xaxb
ln x2 5
ln x2 5
C
C
1 xaxb
A xa
B xb
41 a 5b 2c 91 12a 4b 3c
1 Ax b Bx a 1 Ax Ab Bx Ba 1 xA B Ab Ba AB 0 Ab Ba 1 .... Ba Bb 1 Ba b 1 1 1 B ab , A ba I I
1 ba
xa 1 lnx ba
1 ab
xb
a
43 4b 3c c 43 43 b 3 4 12a 4b 3 43 b 12a 43 91 3 3 55 7 4 b b2 4 b 43 3 3 3 3 2 47c 3c b 7, c 5, a 4 dx dx dx 7 x3 5 x4 4 lnx 1 7 lnx 3 5 lnx 4 C I 4 x1 1284
1 ba
1 ab
dx xa
1 ab
dx xb
lnx b C
5
1281 I
x 2 5x9 x 2 5x6
1 x2x3
dx dx 3 a x2
b x3
dx x 2 5x6
x 3
dx x2x3
|x 2x 3
1 ax 3 bx 2 ax 3a bx 2b xa b 3a 2b a b 0. . . a b 3a 2b 1 3b 2b 1 b 1, a 1 1 1 I x x2 x3 x 3 lnx 2 3 lnx 3 C 1282
dx dx dx dx 121 x1 13 x2 14 x3 x1x2x3 121 lnx 1 13 lnx 2 14 lnx 3 b c 1 a x1 x2 x3 |x 1x x1x2x3
2x 3
1 ax 2x 3 bx 1x 3 cx 1x 2 1 ax 2 5ax 6a bx 2 2bx 3b cx 2 cx 2c 1 x 2 a b c x5a 2b c 6a 3b 2c abc 0 5a 2b c 0 6a 3b 2c 1 b 13 , a 121 , c 14 1283
2x 2 41x91 dx x1x3x4 2x 2 41x91 a x1 x1x3x4 2
c x4
|x 1x 3x 4
2x 41x 91 ax 3x 4 bx 1x 4 cx 1x 3 2x 2 41x 91 ax 2 ax 12a bx 2 5bx 4b cx 2 2cx 3c 2x 2 41x 91 x 2 a b c xa 5b 2c 12a 4b 3c 2 abc
dx
5x 3 2 xx1x4
dx 5 dx
xx1x4 b ax x1
25x 2 20x2 xx1x4 2
c x4
dx 5
x 3 25 xx1x4
dx
25x 2 20x2 xx1x4
|xx 1x 4
2
25x 20x 2 ax 5x 4 bxx 4 cxx 1 25x 2 20x 2 ax 2 5ax 4a bx 2 4bx cx 2 cx 25x 2 20x 2 x 2 a b c x5a 4b c 4a 25 a b c 20 5a 4b c 2 4a a 12 4a 3b 4 12 3b 2 3b 5 b 73 1 73 c 25 2 c 161 6 dx dx x4 I 1 12 dxx 73 x1 161 12 ln x 73 lnx 1 6 7 1 lnx 4 C I 5x I 1 5x 2 ln x 3 lnx 1 161 6
161 6
lnx 4
1285
dx xx1 2
x
1 t
dx dt t2
b x3
3 2 5x 3 2 dx xx5x2 5x4 x 3 5x 2 4x 2 3 2 2 x 5x 4x 5 5x 4x
1 t
1 t
dt t2 2
1 dt 1t
dt 2 t 1t t dt 1t 2
t 1t 2
dt
ln1 t
1 1t
t11 1t 2
dt
ln x1 x
x x1
C
1286
x 3 1 dx 4x 3 x x 1 4 4
3 1 4x 3 4 dx 14 4x 4xxx4 dx 14 3 x 4 4x 3 x x4 x 1 x4 dx 4 4 x2x12x1 dx x4x 2 1 b c x4 a x 2x1 2x1 |x2x 12x x2x12x1
dx
1
1 4
x4 4x 3 x
dx
x 2 8x7 x2 2 x5 2 2
x 4 a2x 12x 1 bx2x 1 cx2x 1 x 4 4ax 2 a 2bx 2 bx 2cx 2 cx x 4 x 2 4a 2b 2c xb c a a 4 4a 2b 2c 0 bc 1 16 21 c 2c 0 c 92 b 1 c 1 92 72 I 4x dx
72
dx 2x1
92
2x1
7 4
dx 4 ln x
x 4 6x 3 12x 2 6 x 3 6x 2 12x8 3 2
dx
x 4 6x 3 12x 2 6 x2 3 3 2
ln2x 1
9 4
ln2x 1 C
x 4 6x 3 12x 2 8x 6 8x I xdx 2 4x33 dx
x2 2
8
x2 dx 22 x2 2
x2 2
dx x2 3
2
4x838 x2 3 8 x2 x2 2
dx
x2 2
11 x2 2
5x 2 6x9 2 x 2 2x3 2
ax 2 2x32x2axb
8
x2 x2 3
dx 22
dx x2 3
C
x 2 2x3 2
2
2
|x 2x 3
x 2x3
1289
x 2 8x7 2 x 2 3x10
x2
27 49
dx
x2 2
2x3 3 x 2 3x2
dx
dt t3
2t12
1 2 2x 2 3x2
x 2 8x7 x2 2 x5 2
dx
|x 2 2 x 5 2
8 dx 49
x5 2
27 49x2
C
1291
dx
1 2
I x I 1 x ln x
dx xx 2 1
1
1 ax a bx cx ab 0 c 0a 1b 1 I 1 dxx x 2x1 dx ln x
2
C
30 343
x5
1 2
lnx 2 1
2
lnx 1 C
1292
x4 dx x 4 1 1 x 2 1x 2 1 3
dx x
x 4 11 dx x 4 1 axb cxd x 2 1 x 2 1 2
2
2
dx x 4 1
x I1
|x 1x 1
1 ax ax bx b cx 3 cx dx 2 d 1 x 3 a c x 2 b d xa c b d ac 0 a c bd 0 b d ac 0 2c 0 c 0a 0 bd 1 2d 1 d 12 b 12 I1 Ix
1 2
dx
12
dx
x 2 1 x 2 1 12 arctanh x 12
1 2
arctanh x
arctan x C
1293 dx
d x5 2 2
dx
3 x 3 x1 dx x xx1 3 x dx xx 2 1 bxc 1 a x x 2 1 |xx 2 xx 2 1 2 2
5x 6x 9 ax 2x 3 2x 2ax b cx dx 2x 3 5x 2 6x 9 ax 2 3a 2xb 2b cx 3 2cx 2 3cx dx 2 2dx 3d 5x 2 6x 9 x 3 c x 2 a 2c d x2b 3c 2d 3a 2b 3d c 0 a 2c d 5 d a. . . d 5 a 2b 3c 2d 6 2d 2b 9 3a 2b 3d
5x3 x 2 2x3
30 dx 343
2
15 2d 2b 3a 2b 3d 5d 3a 55 a 3a 8a 25 15 a 5, d 0 6 2b. . . b 3 2 0 x 2 2x3 I 5x2 6x92 dx x 5x3 dx 2 2x3
c x5
2x 3dx dt
dx|
cxd x 2 2x3
b x2 2 2
x 2 3x 2 t
I
1290
68x x 3 6x 2 12x8
1288 5x 2 6x9 dx x3 2 x1 2 5x 2 6x9 axb cxd I 2 dx x 2 2x3 x 2 2x3 x 2 2x3 axb 5x 2 6x9 x 2cxd 2 2x3 x 2 2x3 x 2 2x3
I
dx
x 4 6x 12x 6 : x 6x 12x 8 x
a x2 3
x 8x 7 ax 8ax 5ax 50a bx 10bx 25b cx 3 cx 2 16cx 20c dx 2 x 2 8x 7 x 3 a c x 2 8a b c d x5a 10b 16c 4d 50a 25b 20c ac 0 8a b c d 1 5a 10b 16c 4d 8 50a 25b 20c 4d 7 30 30 8 a 343 , c 343 , b 27 , d 49 49
1287 I
dx x 2 4x3x 2 4x5
dx x3x1x 2 4x5
1 2
arctan x
30 343
lnx 2
8 49x5
30 343
ln
4 1612 2 42 3 2 42 1 2 1 x3x1x 2 4x5 3 2
x 12 x1 x2
a x3
b x1
3
cxd x 2 4x5 2
1
1
1
0
0
3 1
3
1
4
1
0
1
7
3
4 0
5 15
0
3
1
0
0
1
0 2 7
0 2
0
2
1
1
1
1
0 0
1 35
15
1
1
0
0
0
2
7
1
0
0
8
2
4 0
:4
1294
0 10
5
3
: 5
1
0
0 2 7
0 2 0
0
1 2 12
0 0 3 5
0
1
1
0
0
0
0 2 7
1
0
1 2 1 2
1
0
85
15
1
0
85
15
0
2
0
0
1
1
1
0
0
1
1
1
0
0
0 2
7
1
0
0 2
7
1
0
2
0
85
15
1 2
7
0
0
12
1
1
1
0
0
0 2
7
1
0
0
0
152
2
0
0
0
15 2
815 5
3
0
0
0 2
7
1
0
0
152
0 d
0
2
1
0
152
2
0
0
0
12
85
15
1
0
0
0 2
7
1
0
2
0
0
0
152
0
0
15 2
0
26 3
dx x3
I
1 52
lnx 3
1 20
dx dz
2 C
|x 1x 2 x 1
1 0 0
1 1 0 0
0 2 1 0 2
0 1 2 1
1
1
0
0
0
2
1
0
0 2 4 2
1 1
0
0
0 2
1
0
0 0 3 2
15 b 13 , c I
0 1 1
1 3
x1
dx
2 , a 13 3 13 x 23
lnx 1
I I
1 3
lnx 1
I I
1 3
lnx 1 lnx 1
dx 13 lnx 1 13 x 2x2 x 2 x1 x1 1 6 x 2x4 2 x1 dx 1 16 x 2x1 2 x1 2 x 2 x1 dx 16 lnx 2 x 1 12 2 x 12 14 1
1 3 1 3 1 3
lnx 1
1 6 1 6
lnx 2 x 1 lnx 2 x 1
1 2
3 3
dx x 12
2
arctan
34 2x1 3
C
1295
, b 201 , a
1 52
dx x 2 4x5
dx x1xdx2 x1 x 3 1 1 a x1 x 2bxc x1 x1x 2 x1 2 2
I
3 26
I
24 3
7 1 2 65
x2 z
1
0
1
dx
x 2 4x5
dt 7 1 dx 7 dz 1 1 65 ln t 130 t 2 65 65 z 2 1 x2 2 1 7 7 1 1 2 ln t 130 arctan z 65 lnx 4x 5 130 arctanx 2 65 7 1 1 1 2 lnx 3 lnx 1 lnx 4x 5 arctanx 52 20 65 130
1
0
0
1 1 1 0
0
0
x 2 4x5
2x4 x 2 4x5
2x 15 2
1 ax bx ax bx cx c a 1 x 2 a b xa b c c a ab 0 a b c 0 ca 1
15
0
1 2 12
1
1 1 1
1
0
1
1
0
1
I1 I
1
1
1
0
1 65
2x 4dx dt I1
1
0 2 7 0 2
1
1
1 2
2x4 72
1 65
lnx 1
x 2 4x 5 t
|x 3x 1x 2 4x 5
5
1 20
lnx 3
1 65
I1
1 ax 3ax ax 5a bx bx 7bx 15b cx 3 4cx 2 3cx dx 2 4dx 3d 1 x 3 a b c x 2 3a b 4c d xa 7b 3c 4d 5a 15b 3d
1
1 52
I
dx x1
1 20
1 52
,c
2 65
2 x 3 65 26 x 2 4x5
lnx 1
dx
1 65
dx x 4 1
dx 2 x 2 1 2x 2 11
1 2x
3 26
65
x 2 4x5
dx
x 2 2 x1
x 2 2 x1
dx dx 2 x 2 2 x1 x 2 2 x1 x 2 1 2x 2 axb 2 cxd x 2 2 x1 x 2 x1
1 ax b x 2 2 x 1 cx d x 2 2 x 1 1 ax 3 ax a 2 x 2 bx 2 b b 2 x cx 3 cx c 2 x 2 dx 2 d d 2 x
1 x 3 a c x 2 a 2 b c 2 d x a b 2 c d 2 ac 0 a 2 bc 2 d 0 ab 2 cd 2 0 bd 1 1
0
1
0
0
2
1
2
1
0
1
2
1
0
1
0
1
0
0
1 0
0
1
0
1 0
0
2 2
0
2
0
:
2
0 2 0
1
1
1
0
0
4 2
1
0
0 0
4 2
2
0 0
4 2
0 1
1 2
d
I
2 4
x 12
I
2 4
I
2 8
2 x1
1
0
1
1 1 0
0 1
2 8
dz z
0
x2
1 4
2 x2
2 8
2 8
I 18 2 ln t
1 8
2 ln z
1 4
1 4
dx x 4 x 2 1
ln z
2 4
0
0
1 0 0
1
0
0
0
1
0
0
0 1 0
0
1
0
0
0 0 1
0 0 1
0
0
1
0
1 0 0 1
0
0
1
0 1 0
0
1
0
0 0 1
0
0
1 2
2x2 2
2 8
x 2 2 x1
2 2 x1 dx x 2 2 x1 dx x
2 2
2
2 8
2 8
12 1
2 8
x 2 1 x 4 x 2 1x 2 1
dx
dx
1
0
0
0
1
0
0 0 0
0
2
0
0
1 1
0 0 0
0
0
2 1
gauss
dx 2
x 12
I lnx x I ln
2
2 4
x 2 1 x 3 1x 3 1
dx
x 2 1 x 3 1
1 2
x 2 1 x 3 1
34
x 2 x1 x 2 x1
2 3 3
dz z
x 2 1 x1x 2 x1 x1x1 dx x1x 2 x1 1 2
dx
1 2
x 2 1 x1x 2 x1
dx
dx x 12
2
14 1
2x1 3 3
arctan
ln z
2x1 3 1 arctan lnx 2 3 2 3 2 3 2x1 arctan 3 arctan 2x1 3 3 3 2 3 3
dx x 12
2
34
x 1
2 3 3
arctan
2x1 3 3
C
1297 dx 2 2
arctan 2 4
?????????
x2
2 x1 dx x 2 2 x1 1 4 x
1 2
I ln t
x 2 2 x1
x 2 1
x1 dx 12 x 3 1 dx 2x11 dx 12 x 2 x1 x 2 x1 dx 2x1 dx x 2 x1 2x11 dx x 2 x1 x 2 x1 dt dx dx 2x1 dx x 2 x1 2 2 t x x1 x 12 14 1
2
ln x 2 2 x 1 x 2 1 x 6 1
0
x 3 1
I lnx x 1
x2
0
0 0 0 2
12 x 2 12
1 2
I
2x2 2
x 2 12 x 3 1
2
arctan 2 x 1
ln x 2 2 x 1
0
0
I
2 x2 x 2 2 x1
1296
1
0 1 0
I
x 2 2 x1
|x 3 1x 3 1
1 0 0
2
I
~
2 x1
2 8
2 8
0
I
x 2 2 x1
2 ln z
ln t
1
2
2x 2 2
1 8
2 8 2 8
0
2 1
x 2
I 18 2 ln t
I
1
2 4
2x 2 dx dz
0
dx 2 exf x 3 1 3
e0 d 12 f 12 1 c 2 0 c 12 b 0 1 a 2 0 a 12
2x 2 dx dt dt t
0
~
0 0
x 2x1 z
0
ax 2 bxc x 3 1
x 1 ax bx cx 1 dx 2 ex fx 3 1 x 2 1 ax 5 ax 2 bx 4 bx cx 3 c dx 5 dx 2 ex 4 ex fx 3 f x 2 1 x 5 a d x 4 b e x 3 c f x 2 a d xb e c f ad 0 be 0 cf 0 ad 1 be 0 c f 1
1 1
0
x 2x1 t 2 8
4 2
2 0
2
I
1
0 0
x 12
0
1 0
2 4
2 8
0
4 2
,a
0
1
x 2 1 x 3 1x 3 1 2
0
2
4 2
1 2
2 4
0
1
2 2
0
1 2 1
0 1
0 0
1
1
2 2
0 1
x 2 2 x1
x 2 2 x1 2x 2 2 x2
~
,b
x 2
~
2
1 0
0
x 2 2 x1
2 2
1
0 1 2 4
,c
1
1
0
1
: 2 ~
2 0
1
2 2
1
b d
2
12 1
dx 2 1x 2
x tan t
2x1
arctan 2 x 1
2 4
dx 1 tan 2 tdt
arctan
1tan 2 tdt
1 2
1tan 2 t
2
dt 1tan 2 t
1 cos 2t
1 2
dt
dt 1 1 2
sin 2 t cos 2 t
dt cos 2 tsin 2 t cos 2 t
cos 2tdt
1 2
t
1 4
cos 2 tdt cos 2 tsin 2 t
sin 2t
1 2
cos 2 tdt
cos 2 tdt 1
arctan x
1 4
sin2 arctan x
I1
1298 I
dx
3x5 x1 2 1
dx 3
I 3
dx 5 arctanx 1
I 3
x1 x1 2 1
dx 3 arctanx 1 5 arctanx
x1 t
3
dx dt
t t 2 1
t2 1 u
dt
2tdt du
dx x 5 3 dx x1 2 1 x1 2 1 dx dx 3 5 arctanx 1 x1 2 1 x1 1 3 dx 2 arctanx 1 x1 2 1
x dx x1 2 1 x1 3 x1 2 1
x11 x1 2 1
I1 3 2
3x5 x 2 2x2
3
du 2u
3 2
ln u
lnt 2 1 I 1 32 ln x 1 2 1 C I 32 ln x 1 2 1 2 arctanx 1 C
I1 I1 I2 I2 I2
1299
x
1 2
1 2 2 x1x x1 2
I2 a x1
bxc 2 x 2 x1
dxe x 2 x1
|x 1x 2 x 1
2
I2
2
1 ax x 1 bx cx 1 dx ex 1x 2 x 1 1 ax 4 2ax 3 3ax 2 2ax a bx 2 bx cx c dx 4 2dx 3 2dx 2 dx ex 3 2e x 2 2ex e 1 x 4 a d x 3 2a e 2d x 2 3a b 2d 2e x2a b c d 2e a c ad 0 2a e 2d 0 3a b 2d 2e 0 2a b c d 2e 0 ace 1 1 0 0 1 0 0
1 0 0
2 0 0 2 1 0
3 1 0 2 2 0 2 1 1 1 2 0
Gaussova eliminacija
lnx 1
dx x 2 x1
lnx 1
dx x 2 x1 14 14
I lnx 1 I 1 I 2
0 0 1
xdx 2 x 2 x1 dx 2 x 2 x1
dx 2 x 2 x1
xdx x 2 x1
0
0 0
0 0 0 1 1 1 0 0 0
x1x x1
0 0
0 1 0 1 2 0
1 0 1 0 1 1 e 0d 1c 0 b d 2e 0 b1 0 b 1a 1 dx dx x1 2 2
1
0
1 0
I2 I2
dx
2 3
dx 2 x 2 x1
x
1 2
34
dx dt dt t 2 34
2
x 12
t
2 3
arctan
t
2 3
2 3
arctan
x
1 2
2 3
2x1 3
arctan
dx x 12
2
2
34
t
dx dt dt 2
t 2 34
u
dx 2 x1x 2 x1
dx x 2 x1 14 14
256t 3 4t 2 3
256t 2 3 4t 2 3
2
64
dt 2 4t 2 3
2
I2 3
t t 2 34
dt du
t t 2 34
dt dv
2
2
t t 2 34
1 t 2 34
tv dt 3 dt
t 2 34
3
t 2
t 2 34 dt t 2 34
3
64
4t 2 33 3 4t 2 3
t t 2 34
. . . . odusta san
2
dt dt t 2 34
2
3
dt t 2 34
3
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