1031 5a2x6dx 5a2 x6dx 5a2 x77 5
7 a2x7 C
10326x2 8x 3dx 6 x2dx 8 xdx 3 dx 6 x33 8 x22 3x 2x3 4x2 3x C
1033 xx ax bdx xx2 xb ax ab x3 x2b ax2 xabdx x3dx b x2dx a x2dx ab xdx x4
4 b x3
3 a x3
3 ab x2
2
14 x
4 13 bx
3 13 ax
3 12 abx
2 C
1034a bx32dx a2 2abx3 b2x6dx a2x 2ab x44 b2 x
6
7
a2x 12 abx
4 17 b
2x6 C
1035 2px dx
2px t|
2pdx dt
t dt2p 12p t
12 dt 1
2p23 t
32 1
3p t32 1
3p 2px32 2
3 2 x px C
1036
dxn x
x 1n dx x1n
nn
1n nn x
n1n
n1n
n xn1n
n1 C
1037nx 1nn dx
nx u
ndx du
u 1nn du
n 1n u
1nn du 1
nu1nn nn
1nn nn
1nu1n
1n
u 1n nx
1n C
1038a 2
3 x23 3dx a2 3a 4
3 x23 3a 2
3 x43 x2 dx
a2x 3a 43 x
5353 3a 2
3 x7373 x3
3 a2x 95 a
43 x
53 9
7 a23 x
73 1
3 x3 C
1039
x 1x x 1dx x 32 1 dx x 2
5 x52 C
1040
x21x223 x2
dx x4x22x23dx x 10
3 x43 2x 23 dx 3
13 x133 3
7 x73 2 3x 1
3
313 x
133 3
7 x73 6 3 x C
1041
xmxn 2
xdx x2m2xmnx2n
xdx x2mx 12 2xmnx 12 x2nx 12 dx
x2m 12 2xmn 12 x2n 12 dx x2m12 1
2m 12 1 2 xmn
12 1
mn 12 1 x2n
12 1
2n 12 1 C
x2m12
2m 12 2 x
mn 12
mn 12 x2n
12
2n 12 C
1042
a x 4
axdx ax 12 a 2 a x x2dx ax 12 a2 4a 3
2 x 6ax 4 a x32
a12 x 12 a2 4a 3
2 x 6ax 4 a x32 x2 a
12 1
xa2 4a 3
2 6 x a 4x a
a12 x
12 1
12 1a2 4a 3
2 x 6a 23 x32 4 x22 a x
5252
a12 2a2 x 4a 3
2 x 4ax 32 2x
2a 32 x 4ax 4 a x
32 2x2 2
5 ax52 C
1043 dxx27
17arctan 1
7x C
1044 dxx210
110arctanh 1
10x C
1045 dx
4x2 arcsinh 1
2 x C
1046 dx
8x2 arcsinh 1
4 2 x C
1047
2x2 2x2
4x2dx 2x2 2x2
2x2 2x2dx 1
2x2dx 1
2x2dx
arcsin 12 2 x arcsinh 1
2 2 x C
1048*a tan2xdx sin2x
cos2xdx 1cos2x
cos2xdx 1
cos2xdx dx 1
cosx sinx x C
1048*b tanh2xdx sinh2
cosh2 1cosh2x
cosh2 dx
cosh2 dx sinhx
coshx x tanhx x C
1049a cot2xdx cos2x
sin2xdx 1sin2x
sin2xdx dx
sin2x dx cotx x C
1049b coth2xdx cosh2x
sinh2xdx 1sinh2x
sinh2xdx dx
sinh2x dx cothx x C
1050 3xexdx 3exdx
3ex u3xex1 ln3dx du
3ex du3xex1ln3 du
1ln3 11ln3 u
3ex
1ln3 C
1051** adxax a dx
ax
a x u|
dx du
a duu a lnu a lna x C
1052** 2x32x1 dx
2x122x1 dx dx
22x1 dx x
22x1 dx
2x 1 u|
2dx du
x duu x lnu x ln2x 1 C
1054 xdxabx 1
b bxdxabx 1
b bxaa
abx dx 1b bxaa
abx dx 1b dx a
b dxabx
xb a
b dxabx
a bx u|
bdx du
xb a
b
dubu x
b a
b2 duu x
b a
b2lnu x
b a
b2lna bx C
1055
axbx a xba
x dx a xdxx b
x dx
a x
x dx b dxx a
xx dx b
dxx
a dx a
x dx b
dxx ax
a dx
x b dxx
ax b a
dxx ax
ba dx
x
x u|
dx du
ax ba
duu ax
ba2 duu ax
ba2
lnu a x ba2
lnx C
1056 x21x1
x2 1 : x 1 x 1 2x1
x2 x
1 x
x 1
2 xdx dx 2 dx
x1 x2
2 x 2 lnx 1 C
1057 x25x7
x3 dx x2
x3 dx 5 xx3 dx 7
dxx3 x299x3 dx 5 x33x3 dx 7 lnx 3
x 3dx 9 dxx3 5 dx 15
dxx3 7 lnx 3
x2
2 3x 9 lnx 3 5x 15 lnx 3 7 lnx 3 1
2 x2 2x lnx 3 C
1058 x4x21
x1 dx x4
x1 dx x2
x1 dx dxx1 x411x1 dx x211x1 dx lnx 1
x41x1 dx lnx 1 x21x1 dx lnx 1 lnx 1
3 lnx 1 x21x1x1x1 dx x1x1
x1 dx
3 lnx 1 x 1dx x2 1x 1dx 3 lnx 1 x2
2 x x3 x2 x 1dx 3 lnx 1 x2
2 x x4
4 x3
3 x2
2 x
14 x
4 13 x
3 x2 2x 3 lnx 1 C
1059a b
xa 2dx a2dx 2 ba
xa dx b2
xa2dx a2x 2ab lnx a b2 dx
xa2
x a t|
dx dt
a2x 2ab lnx a b2 dtt2
xa2 2ab lnx a b2t xa2 2ab lnx a b2
xa C
1060* xx12
dx x11x12
dx dxx1
dxx12
lnx 1 dxx12
x 1 t|
dx dt
lnx 1 dtt2 lnx 1 1
t lnx 1 1x1 C
1061 bdy
1y b dy
1y
1 y t|
dy dt
b dtt 2b t 2b 1 y C
1062 a bx dx
a bx t|
bdx dt
t dtb 1
b t dt 2
3b t32 2
3b a bx32 C
1063** x
x21dx
x2 1 t|
2xdx dt
xtdt2x 1
2 dtt t x2 1 C
1064
x lnxx dx 1
xdx 1x lnxdx
lnx t| dxx dt
2 x tdt 2 x 12 t2 2 x 1
2 ln2x C
1065
dx3x25
13
dxx2 53
13
155 arctan 15
5 x 1515 arctan
155 x C
1066
dx7x28
17
dxx2 87
17
14 14 arctanh 1
4 x 14 1428 arctanh
144 x C
1067 dxababx2
1ab
dxabab x
2 1
ab1abab
arctanh xabab
ab
ab abarctanh x ab
ab
ab
ab 1
a2b2arctanh x a2b2
ab C
1068
x2
x22dx x222
x22dx dx 2 dx
x22 x 2 arctan 2
2 x C
1069
x3
a2x2dx x3
axax dx x3a3a3axax dx
a3x3axax dx
a3
a2x2dx
axa2axx2axax dx a3 1a arctanh x
a a2axx2ax
a3 1a arctanh xa
a3 1a arctanh xa a2axx2
axdx a3 1a arctanh x
a a2ax dx
xxaax
dx
a3 1a arctanh xa a2 lna x xdx 1
2 x2 a2 lna x a2 arctanh x
a C
1070 x25x6
x24dx x25x42
x24dx dx 5 xdx
x24 2 dx
x24
x 52 lnx
2 4 arctan 12 x C
1071
dx
78x2 1
8 dx
78 x
2 1
2 222arcsinh 2 14
7 x 24 arcsinh
2 147 x C
1072
dx
75x2 5
5 dx75 x
2 5
5 arcsin357 x C
1073
2x53x22
dx 13
2x5x2 23
dx 13
2xx2 23
dx 5 dxx2 23
x2 23 t|
2xdx dt
13
dtt 5
62 arctanh
62 x 1
3 ln t 56 6 arctanh 1
2 x 6
13 lnx
2 23
56 6 arctanh 1
2 x 6 C
1074
32x5x27
dx 15
32xx2 75
dx 35
dxx2 75
15
2xdxx2 75
3 3535 arctan 35
7 x 15
2xdxx2 75
x2 75 t|
2xdx dt
3 3535 arctan 35
7 x 15
dtt 3 35
35 arctan 357 x
15 ln t
3 3535 arctan 35
7 x 15 lnx
2 75 C
1075
3x15x21
dx 55
3x1x2 15
dx 3 55 x
x2 15dx 5
5 dx
x2 15
x2 15 t|
2xdx dt
3 55
dt2
t 5
5 arcsinh 5 x 3 510
dtt 5
5 arcsinh 5 x
3 510 2 t 5
5 arcsinh 5 x 3 55 x2 1
5 55 arcsinhx 5 C
1076 x3
x24dx xdx
x24 3 dx
x24
x2 4 t|
2xdx dt
dt2
t 3 ln x x2 4 1
2 dtt 3 ln x x2 4
12 2 t 3 ln x x2 4 x2 4 3 ln x x2 4 C
1077
xdxx25
x2 5 t|
2xdx dt
dt2t 1
2 dtt 1
2 ln t 12 lnx
2 5 C
1078 xdx2x23
2x2 3 t4xdx dt
dt4t 1
4 dtt 1
4 ln t 14 ln2x
2 3 C
1079 axba2x2b2
dx a 0 1a axb
ax2 b2a
dx 1a ax
ax2 b2a
dx ba2 dx
x2 b2
a2
1a ax
ax2 b2a
dx ba2 dx
x2 b2
a2
ax2 b2a t|
2axdx dt
1a
dt2t dx
ba2 a
barctanx a
b 1
2a dtt
1a arctan a
bx
12a lnax
2 b2a 1
a arctan abx C
1080
xdx
a4x4
x4 t2| 12
x2 t|
2xdx dt
dt2
a4t2 1
2 dt
a4t2 1
2 arctant
a4t2 1
2 arctanx2
a4x4 C
1081 x2
1x6dx
x6 t2| 12
x3 t|
3x2dx dt
dt3
1t2 1
3 dt1t2
13 arctan t
13 arctanx
3 C
1082
x2dx
x61
x6 t2| 12
x3 t|
3x2dx dt
dt3
t21 1
3 dt
t21 1
3 ln t t2 1 13 ln x
3 x6 1 C
1083
arcsinx1x2
dx arcsinx u|
dx
1x2 du u du 2
3 u32 2
3 arcsinx32 C
1084
arctanx2
x24
arctan x2 u|
2dxx24
du
12 udu
u2
4 14 arctan
2 x2 C
1085
x arctan2x14x2
dx x14x2
dx arctan2x14x2
dx
1 4x2 u|
8xdx du
t2dx14x2
dt
du8u t dt2 1
8 lnu 12
23 t
32 1
8 lnu 13 t
32
18 ln1 4x
2 13 arctan
32 2x C
1086 dx
1x2 lnx 1x2
lnx 1 x2 u| dx
x21 du
duu 2 u 2 lnx 1 x2 C
1087 aemxdxmx u|
mdx du
aeu dum am eudu a
m eu a
m emx C
1088 423xdx
2 3x t|
3dx dt4t u|
4t ln4dt du
4t dt3 13 4tdt 13 4t
du4t ln4 1
3 ln4 du 13 ln4 u
4t3 ln4 1
3 ln4 423x C
1089et etdt 2 etet2 dt 2 sinh tdt 2cosh t Cet etdt etdt etdt et et C
1090
e xa e xa 2dx e2 xa dx 2 dx dx
e2xa
2 xa t| 2dxa dt
et a2 dt 2x a2 etdt a
2 etdt 2x a2 etdt
12 ae
2 xa 2x 12 ae
2 xa C
1091
axbx2
axbxdx a2x2axbxb2x
axbxdx ax
bxdx 2dx bxax dx
ab
xdx 2x ba
xdx ab
x
ln ab 2x
ba
x
ln ba C
1092 a2x1
axdx a2x 12 xdx a 12 xdx a 3
2 xdx a 12 xdx au 23 du at2dt32 x u|
12 x t|
32 dx du dx2 dt
23 lna a
u 2lna a
t 2lna
a32 x
3 a12 x C
1093
ex21xdx ex21xdxx2 1 t|
2xdx dt
et dt2 12 etdt 12 e
t
12 ex21 C
1094 x7x2dx
x2 t|
2xdx dt
7t dt2 12 7tdt
12 ln7 7
t 12 ln7 7
x2 C
1095
e1x
x2dx
1x t|
dxx2
dt
etdt et e 1x C
1096 5 x dx
x
x t| dx2 x
dt
5t2dt 2 5tdt 2ln5 5
t 2ln5 5
x C
1097 ex
ex1 dx
ex 1 t|
exdx dt
dtt ln t lnex 1 C
1098 ex a bex dxa bex t|
bexdx dt
t dtb 1
b t dt 2
3b t32 2
3b a bex
32 C
1099
e xa 1 13 e x
a dx
exa 1 t|
1a e
xa dx dt
t 13 adt a t 13 dt 34 at
43 3
4 aexa 1 43 C
1100* dx2x3
2x 3 t|
2x ln2dx dt
dt
2x ln2t 1
ln2 dt2xt
1ln2
dtt3t
1ln2
dtt23t
1ln2
dtt 32
2 94
1ln2
194
arctanh t 3294
23 ln2 arctanh
23 t 1
23 ln2 arctanh
23 2
x 3 1 C
1101 axdx1a2x
ax u
ax lnadx du
dulna1u2
1lna
du1u2
1lna arctanu
1lna arctana
x C
1102 ebx
1e2bxdx
ebx u
bebxdx du
dub
1u2 1
b du1u2
1barctanhu 1
barctanhebx C
1103
et
1e2tdt
et u
etdt du
du
1u2 arcsinu arcsinet C
1104 sina bxdxa bx z
bdx dz
sin z dzb 1
b sin zdz 1
bcos z 1
bcosa bx C
1105
cos x2dx
x2 z
dx2 dz
cos z 2 dz 2 cos zdz 2 sin z 2 sin x2 C
1106cosax sinax2dx sin2ax 2sinaxcosax cos2axdx 1 2sinaxcosaxdx 1 sin2axdx x sin2axdx
2ax z2adx dz
x sin z dz2a x 12a sin zdz x
12a cos z x
12a cos2ax C
1107 cos x dx
x
x zdx2 x
dz
cos z2dz 2 cos zdz 2sin z 2sin x C
1108 sinlgx dxx sin lnxln2
dxx
lnxln2 z|
dxx ln2 dz
sinz ln2dz ln2 sin zdz ln2cos z ln2cos lnxln2 ln2cos lgx C
1109 sin2xdx 1cos2x2 1
2 dx 12 cos2xdx
2x z|
2dx dz
x2
12 cos z
12 dz
x2
14 cos zdz
12 x
14 sin z
12 x
14 sin2x C
1110*
cos2xdx 12 1 cos2x
2dx 12 dx 12 cos2xdx
12 x
12 cos2xdx
2x z|
2dx dz
12 x
12 cos z
dz2 1
2 x 14 cos zdz
12 x
14 sin z
12 x
14 sin2x C
1111 sec2ax bdx dx
cos2axb
ax b t
adx dt
dta
cos2t 1
a dtcos2t
1a tan t 1
a tanax b C
1112 cot2axdx cos2ax
sin2axdx 1sin2ax
sin2axdx dx
sin2ax dx
ax t
adx dt
dta
sin2t x 1
a dtsin2t
x 1a cot t x 1
a cotax x C
1113
dxsin x
a
xa tdxa dt
adtsin t a dt
sin t a ln tant2 a ln tan
xa
2 a lntan x2a C
1114 dx3cos5x
4
5x 4 t
5dx dt
dt5
3cos t 115
dtcos t
115 ln tan
t2
4
115 lntan
12 5x
4
14
115 ln tan
52 x
18 C
1115 dxsinaxb
ax b t
adx dt
dta
sin t 1a dt
sin t 1a ln tan t
2 1a ln tan axb
2 C
1116 xdxcos2x2
x2 t
2xdx dt
dt2
cos2t 1
2 dtcos2t
12 tan t
12 tanx
2 C
1117 x sin1 x2dx
1 x2 t2xdx dt
sin t dt2 12 sin tdt 12 cos t
12 cos1 x
2 C
1118
1sinx 2
12dx 1
sin2x 2 2
sinx 2 1 dx x 2 dx
sinx 2 dx
sin2x 2
x 2 t
2 dx dt
x 2 dt2
sin t dt2
sin2t x 2 dt
sin t 22
dtsin2t
x 2 ln tan t2
22 cot t
x 2 ln tan x 22 2
2 cotx 2 C
1119
tanxdx sinxcosx dx 1cos2x1cos2x
dx 1cos2x1cos2x
1cos2x1cos2x
dx
1cos22x1cos2x dx
sin22x1cos2x dx
sin2x1cos2x dx
1 cos2x t|
2sin2xdx dt
dt2t dx
12 ln t
12 ln1 cos2x C
1120
cotxdx cosxsinx dx 12 1cos2x12 1cos2x
dx 1cos2x1cos2x
dx 1cos2x1cos2x
1cos2x1cos2x
dx 1cos22x1cos2x
sin2x1cos2x dx
1 cos2x t|
2sin2xdx dt
dt2t 1
2 ln t 12 ln1 cos2x C
1121 cot x
ab dx
xab u|
dxab du
cotua bdu a b cotudu a b lnsinu a b ln sin xab C
1122 dxtan x
5 cot x5 dx
x5 u|
dx5 du
cotu5du 5 cosusinu du
sinu t|
cosudu dt
5 dtt 5 ln t 5 lnsinu 5 ln sin x5 C
1123 tan x dx
x
x u| dx2 x
du
2 tanudu 2 tanu 2 sinucosu du
cosu t|
sinudu dt
2 dtt 2 ln t 2 lncosu 2 ln cos x C
1124 xcotx2 1dxx2 1 u2xdx du
cotu du2 12 cotudu
12 lnsinu
12 ln sinx
2 1 C
1125 dxsinxcosx dx
sin2x2
2 dxsin2x
2x u2dx du
2 du2sinu du
sinu ln tan u2 lntanx C
1126 cos x2 sin
x2 dx
x2 udx2 du
2 cosu sinudu sinu z
cosudu dz
2 zdz z2 sin2u sin2 x2 C1127 sin36xcos6xdx
6x t6dx dt
sin3tcos t dt6 16 sin
3tcos tdt
sin t zcos tdt dz
16 z3dz
124 z
4 124 sin t
4 124 sin
46x C
1128 cosaxsin5ax
dx
ax t
adx dt
1a cos tsin5t
dt
sin t zcos tdt dz
1a dzz5 1
4az4 1
4a sin4t 1
4a sin4ax C
1129 sin3x3cos3x dx
3 cos3x z|
3sin3xdx dz
sin3xz dz3sin3x 13
dzz 1
3 ln z 13 ln3 cos3x C
1130
sinxcosxcos2xsin2x
dx sin2x2
cos2xdx 1
2 sin2xcos2x
dx
cos2x t|
2sin2xdx dt
12
dt2
t 14
dtt 1
2 t 12 cos2x C
1131
1 3cos2x sin2xdx 2 sinxcosx 1 3cos2x dxcosx t|
sinxdx dt
2 t 1 3t2 dt1 3t2 z6tdt dz
26 z dz 29 z
32 2
9 1 3t2
32 2
9 1 3cos2x
32 C
1132
tan3 x3 sec2x3 dx
sin3 x3cos3 x3
1cos2 x3
dx sin3 13 x
cos5 13 xdx
cos x3 t|
13 sinx3 dt
3 sin2 x3 dt
t5 3 1t2
t5dt 3 1
t5dt 3 1
t3dt
3 14t4 3 1
2t2 3
4cos4 x3 3
2cos2 x3 C
1133
tanxcos2x
dx
tanx tdxcos2x
dt
t dt 23 t
32 2
3 tan32 x C
1134
a cot23 x
sin2xdx
cos23 x
sin23 x
sin2xdx cos
23 x
sin83 xdx
sin 53 x t|
53 cos
23 xdx dt
cos23 x
sin83 x
dt
53 cos
23 x
35
dt
sin83 x
35
dt
sin53 x sinx
35
dtt sinx 3
5 dt
tt35 3
5 dt
t85
35 5
3t35
t35 sin 5
3 x 35 1
sinx C
b cot23 x
sin2xdx
cotx t|
dxsin2x
dt
t 23 dt 35 t
53 3
5 cotx53 C
1135 1sin3xcos23x
dxcos23x
sin3xcos23x
3x t3dx dt
cos3x z3sin3xdx dz
dt3
cos2t
dz3
z2 1
3 dtcos2t
13 z2dz
13 cot t
13z 1
3 cot3x 1
3cos3x C
1136
cosaxsinax2
sinax dx 12sinaxcosaxsinax dx dxsinax 2 cosaxdx
ax t
adx dt
1a dt
sin t 2a cos tdt 1
a ln tan t2
2a sin t 1
a ln tan ax2 2
a sinax C
1137
csc23xbacot3x dx
dx
sin23xbacot3x
cot3x t 3dxsin2x
dt
dt3
bat 13
dtbat
13a lnb at
13a lnb acot3x C
1138
2sinh5x 3cosh5xdx 2 sinh5xdx 3 cosh5xdx5x t5dx dt
25 sinh tdt
35 cosh tdt 2
5 cosh t 35 sinh t
25 cosh5x
35 sinh5x C
1139
sh2xdx exex2 2dx 1
4 e2x 1
2 14e2x
dx
2x t2dx dt
14 et
dt2
12 x
14
dt2et
18 e
t 12 x
18et
18 e
2x 12 x
18e2x
C
1140 dxsinhx 2
exex dx 2
e2x1ex
dx 2exe2x1
dx
ex t
exdx dt
2dtt21
2arctanh t 2arctanhex C
1141 dxcoshx 2
exex dx 2 dxe2x1ex
2 exdxe2x1
ex t
exdx dt
2 dtt21
2arctan t 2arctanex C
1142 dxsinhxcoshx dx
12 cosh x
2 112 cosh x
2 1 dx
14 cosh2 x2 1
dx12 sinh
x2 2 dx
sinh x2
x2 udx2 du
dusinhu 2arctanheu 2arctanhe x
2 C
1143 tanhxdx sinhxcoshx dx
coshx tsinhxdx dt
dtt ln t ln cosh t C
1144
cothxdx coshxsinhx dx
sinhx tcoshxdx dt
dtt ln t ln sinhx C
1145
x 5 5 x2 dx5 x2 t2xdx dt
5 t dt2 12 5 t dt 512 t
65 5
12 5 x2
65 C
1146 x31x44x1
dx
x4 4x 1 t4x3 4dx dt
dt4t 1
4 ln t 14 lnx
4 4x 1 C
1147 x3
x85dx
x4 t
4x3dx dt
dt4
t25 1
4 dtt25
520 arctan
t 55 5
20 arctanx4 55 C
1148 xex2dx
x2 u
2xdx du
eu2 du 12 eudu
12 e
u 12 ex2 C
1149
3 23x2
23x2dx
2 3x2 t2
3xdx tdt
3tt2
tdt3x 1
3 3t
t2 t223
tdt 33
3tt t22
dt 33
3t t22
33
t
t t22 3 1
t t22
t2 2 z2
tdt zdz
13 3 ln t t2 2 3 1tz zdzt 13 3 ln t t2 2 3 1
z22zzdz
13 3 ln t t2 2 3 1z22
dz 13 3 ln t t2 2 12 3 2 arctan 1
2
13 3 ln t t2 2 12 3 2 arctan 2
2 t2 2
13 3 ln 2 3x2 2 3x2 2 12 3 2 arctan 2
2 2 3x2 2
13 3 ln 2 3x2 3 x 1
2 3 2 arctan 12 2 3 x C
3 23x2
23x2dx 3
23x2dx dx
23x2 dx
x2 23 3
3 dx
x2 23
12 6 arctan 1
2 x 6 33 arcsinh 1
2 x 6 12 6 arctan 1
2 x 6 13 3 arcsinh 1
2 x 6 C
1150
x31x1 dx x
2dx xdx dx 2 dxx1
x3 1 : x 1 x2 x 1 2x1
x3 x2
1 x2x2 x1 xx 12
13 x
3 12 x
2 x 2 lnx 1 C
1151 dx
ex e 12 xdx
12 x u
dx2 du
2 eudu 2eu 2e 12 x C1152 1sinxxcosx dx
x cosx u1 sinxdx du
duu lnu lnx cosx C
1153
tan3xcot3xsin3x dx
3x u3dx du
13
tanucotusinu du 1
3 sinucosu
cosusinu
sinu du 13
1cosu 1
3 cosusin2u
sinu tcosudu dt
13 ln
1sinucosu 1
3 dtt2 1
3 ln1sinucosu 1
3t 13 ln
1sinucosu 1
3sinu 13 ln
1sin3xcos3x 1
3sin3x C
1154 dxx ln2x
lnx tdxx dt
dtt2 1
t 1lnx C
1155
sec2xtan2x2
dx dx
cos2x
tan2x2
tanx tdxcos2x
dt
dt
t22 ln t t2 2 ln tanx tan2x 2 C
1156 2 x
2x21dx2x21
22x21
dx x
2x212dx
2x2 1 t4xdx dt
dxx2 12
dt4t2
2 arctanx 2 14t 2 arctanx 2 1
42x21 C
1157 asinx cosxdx
sinx tcosxdx dt
atdt 1lna a
t 1lna a
sinx C
1158 x2
3 x31dx
x3 1 u|
3x2dx du
du3 3 u 1
23 u2 1
23 x3 12 C
1159
xdx
1x4
x2 u|
2xdx du
du
2 1u2 1
2 arcsinu 12 arcsinx
2 C
1160 tan2axdx
ax u|
adx du
1a tan2udu 1
a sin2u
cos2udu 1
a 1cos2u
cos2udu 1
a ducos2u
1a du 1
a tanu u2
1a tanax 1
2 ax C
1161 sin2 x2 dx
x2 tdx2 dt
2 sin2tdt 2 12 1 cos2t 1 cos2t t 12 sin2t
x2
12 sinx C
1162 sec2xdx
4tan2x
tanx tdxcos2x
dt
dt
4t2 arcsin t
2 arcsin tanx2 C
1163 dxcos xa
xa tdxa dt
a dtcos t a lnsec t tan t a ln
1sin xa
cos xa C
1164
3 1lnxx dx
1 lnx tdxx dt
3 t dt 34
3 t4 34
3 1 lnx4 C
1165
tan x 1 dxx1
x 1 tdx
2 x1 dt
2 tan tdt 2 lncos t 2 ln cos x 1 C
1166 xdxsinx2
x2 t
2xdx dt
12
dtsin t
12 ln
1cos tsin t 1
2 ln1cosx2sinx2
C
1167
earctanxx ln1x2 1
1x2dx earctanx
1x2dx x ln1x
2
1x2dx dx
1x2
arctanx udx1x2
du
ln1 x2 t2xdx1x2
dt
eudu 12 tdt arctanx eu
14 t2 arctanx
earctanx 14 ln
21 x2 arctanx C
1168 sinxcosxsinxcosx dx
sinx 2t1t2
cosx 1t21t2
dx 2dt1t2
2t1t2 1t21t2
2t1t2
1t2
1t2
2dt1t2
2 t22t12t1t2 1t2
dt
t22t12t1t2 1t2
atb2t1t2
ctd1t2
t2 2t 1 at b1 t2 ct d2t 1 t2t2 2t 1 at at3 b bt2 2ct2 ct ct3 2dt d dt2t2 2t 1 t3a c t2b 2c d ta c 2d b d
a c 0b 2c d 1a c 2d 2b d 1a 1,d 0,c 1,b 1
t22t12t1t2 1t2
dt t12t1t2
dt t1t2
dt
2t 1 t2 z21 tdt dz
1 t2 u2tdt du
2 dz2z du2u 2
2 ln z 22 lnu
ln2t 1 t2 ln1 t2 ln2 tan x2 1 tan
2 x2 ln1 tan
2 x2 C
sinxcosxsinxcosx dx sinxcosxsinxcosxsinxcosxsinxcosx dx sin
2x2sinxcosxcos2xsin2xcos2x
dx
12sinxcosxsin2xcos2x
dx 1sin2xcos2x dx 1sin2xcos2x dx
dxcos2x tan2xdx
2x t2dx dt
12 dtcos t
12 tan tdt
12 ln
1sin tcos t
12 lncos t
12 ln
1sin2xcos2x 1
2 lncos2x C
1169
1sin x
2
2
sin x2
dx
x2 t
dx2 dt
2 1sin t2
sin t 2 1sin t2
sin t 2 12sin tsin2tsin t dt 2 dtsin t 2 dt sin t
2 ln 1cos tsin t 2t cos t 2 ln1cos x
2
sin x2
2 x2 cos x
2 C
1170 x2
x22dx x222
x22dx dx 2 dx
x22 x 2 arctanh 1
2 x 2 C
1171
1x2
x1x2 dx x22x1
x1x2 dx x
1x2 dx 2 dx
1x2 dx
x1x2
x1x2
dx 2arctanx dxx1x2
1 x2 t2xdx dt
dt2t 2arctanx dx
x1x2 1
2 ln t 2arctanx dx
x1x2 1
x1x2 a
x bxc1x2
1 a1 x2 xbx c a ax2 bx2 xc1 x2a b xc aa b 0 a 1b 1c 0I1 ax dx bxc
1x2 dx a lnx x
1x2 dx lnx 1
2 ln1 x2
I 12 ln1 x
2 2arctanx lnx 12 ln1 x
2 2arctanx lnx C
1172
esin2x sin2xdx 2 esin2x sinxcosxdxsin2x t
2sinxcosxdx dt
etdt esin2x C1173
53x43x2
dx 543x2
dx 3x43x2
dx 543x2 : 3
: 3 dx 3x43x2
dx
4 3x2 t6xdx dt
53 dx
43 x
2 3
dt6
t 5
3arcsin 1
2 3 x 12
dtt
53 3 arcsin 1
2 3 x t 53 3 arcsin 1
2 3 x 4 3x2 C
1174
dxex1
ex 1 texdx dt
dtext
dtt1t
dtt2t
dt
t 122 14
t 12 u
dt du
duu2 14
12 1
4
ln u 12u 12
ln u 12u 12
ln2u 1 ln2u 1
ln2t 12 1 ln2t
12 1 ln2t 2 ln2t
ln2ex 1 2 ln2ex 1 C1175 dxababx2
1ab dx1x2
1ab arctanx C
1176 ex
e2x1dx
ex t
exdx dt
dt
t21 ln t t2 1 ln ex e2x 1 C
1177
dxsinaxcosax dx
sin2ax2
2 dxsin2ax
2ax t2adx dt
2 dt2asin t
1a dt
sin t 1a ln 1cos t
sin t 1a ln 1cos2ax
sin2ax C
1178
sin 2tT 0dt sin 2tT 0dt 2tT 0 x2T dt dx
T2 sinxdx
12T cosx T
2 cos2tT 0 C
1179 dxx4ln2x
lnx tdxx dt
xdtx4t2
dt4t2
14 ln
2t2t
14 ln
lnx22lnx C
1180
arccosx2
1x2dx
arccos x2 t
12
1 x2
2dx 1
4x2 dx dt
tdt 12 t2 1
2 arccos2 x2 C
1181
etanx sec2xdx etanx 1cos2x
dx
tanx u 1cos2x
dx du
eudu eu etanx C1182 sinxcosx
2sin4xdx
sin2x t2sinxcosxdx dt
dt
2 2t2 1
2 dt
2t2 1
2 arcsin22 t
12 arcsin
22 sin
2x C
1183 dxsin2xcos2x
dxsin22x4
4 dxsin22x
2x u2dx du
4 du2
sin2u 2 du
sin2u 2cotu 2cot2x C
1184 arcsinxx
1x2dx arcsinx
1x2 x
1x2
arcsinx udx
1x2 du
1 x2 t2xdx dt
udu 12
dtt 1
2 u2 t 1
2 arcsin2x 1 x2 C
1185
secx tanxsec2x1
dx 1cosx
sinxcosx
1cos2x
1dx
sinxcos2x
1cos2xcos2x
dx sinxcosx
1cos2xdx sinx
cosx 1cos2xdx
1 cos2x t|2
1 cos2x t2
2 sinxcosxdx 2tdt
sinxcosxt
tdtsinxcosx dt
cos2x dt
t21 arctanh t arctanh 1 cos2x C
1186 cos2x4cos22x
dx
2x t2dx dt
12
cos t4cos2t
12
cos t51cos2t
12
cos t5sin2t
sin t ucos tdt du
12
du5u2
510 arctanh
55 u
510 arctanh
55 sin t
510 arctanh
55 sin2x C
1187*
dx1cos2x
dxsin2xcos2xcos2x :cos2x
:cos2x dx
cos2x
tan2x2 1
cos2xtan2x2dx
tanx tdxcos2x
dt
dtt22
22 arctan
t 22 2
2 arctantanx 22 C
1188
ln x x21
x21dx
ln x x2 1 t
1x x21
1 2x2 x21
dx dt
1x x21
x21 x
x21dx dt
1x21
dx dt
t dt 23 t
32 2
3 ln32 x x2 1 C
1189
x2chx3 3dxx3 3 t3x2dx dt
13 cosh tdt
13 sinh t
13 sinhx
3 3 C
1190 3tanhxcosh2x
dx
tanhx tdx
cosh2x dt
3tdt 1ln3 3
t 1ln3 3
tanhx C
1192 x2x 510dx2x 5 t2dx dt
12 t10dt
122 t
11 122 2x 5
11 C
1191a dx
x x22
x2 2 txdx
x22 dt
dtx22x
x x22 dt
x2 dt
t22 2
2 arctan22 t
22 arctan
22 x2 2 C
dx
x x22
1x t
dxx2
dt
x2dt
x 1t 22
1t dt
2t2
t2
dt
2t2 arcsin 2
2 t arcsin22x C
1191b dxex1
x ln tdx dtt
dtt
eln1t 1
dtt
1t 1
dtt 1t 1
dt1t ln1 t ln1
1ex C
1191c x5x2 37dx5x2 3 t10xdx dt
x5x2 37 dt10x 1
10 t7dt 180 t
8 180 5x
2 38 C
1191d
xdxx1
x 1 tdx
2 x1 dt
2 xdt 2 t2 1dt 23 t3 2t 2
3 x 1 3 2 x 1 C
1191e cosxdx
1sin2x
t sinxdt cosxdx
dt
1t2 arcsinh t arcsinh sinx C
1192 x2x 510dx2x 5 t x t5
2
2dx dt
t52 t10dt2 1
4 t 5t10dt 148 t
12 544 t
11 148 2x 5
12 544 2x 5
11 C
1193 1x1 x
dx
x tdx2 x
dt
1x1 x
2 x dt 2 1t21t tdt 2 tt31t dt
t3 t : t 1 t2 t 2 2t1
t3 t2
t2 tt2 t2t2t 22
2 t2dt tdt 2 dt 2 dtt1 2
3 t3 t2 4t 4 ln1 t
23 x 3 x 2 4 x 4 ln1 x C
1194 dxx 2x1
2x 1 t 2x 1 t2dx2x1
dt x t212
2x1x 2x1
dt dtx dtt212
2 dtt21
2arctanh t 2arctanh 2x 1 C
1195 dx
ex1
ex 1 texdx2 ex1
dt
2 ex1ex
dt
ex1 2 dtex 2 dt
t21 2arctan t 2arctan ex 1 C
1196 ln2xdxx ln4x ln2xdx
xln2xln2
ln2x tdxx t
txdtxtln2 tdt
tln2 tln2ln2tln2 dt
dt ln2 dttln2 t ln2 lnt ln2 ln2x ln2 lnln2x ln2 C
1197
arcsinx2
1x2dx
arcsinx tdx
1x2 dt
t2dt 13 t3 1
3 arcsin3x C
1198
e2x
ex1
ex 1 texdx2 ex1
dt
e2xt2 ex1ex dt 2 t21t
t dt 2 t2 1dt 23 t3 2t
23 ex 1 3 2 ex 1 C
1199 sin3x
cosx dx
cosx t
sinx2 cosx dx dt
2 sin2xdt 2 1 cos2xdt 2 1 t4 dt 25 t5 2t
25 cosx 5 2 cosx C
1200* dx
x 1x2
1 x2 t|2
1 x2 t2
xdx tdt
tdtxxt 1
x2dt 1
t21dt arctanh t arctanh 1 x2 C
1201 x2
1x2dx
x cos tdx sin tdt
cos2t1cos2t
sin tdt cos2t sin tsin t dt cos2tdt 1cos2t2 12 t
14 sin2t
12 arccosx
14 sin2arccosx C
1202 x3
2x2dx
2 x2 cos t2xdx sin tdt
x3
cos tsin tdt2x x2
cos tsin tdt2 2cos t
cos tsin tdt2 2sin tcos t sin t
2 cos tdt
sin tcos t
dt 12 cos t sin tdt
cos t u sin tdt du
duu 1
2 u du 2 u 13 u
32 2 cos t 1
3 cos t32
2 2 x2 13 2 x
232 C
1203
x2a2x dx
x2 a2 sin2txdx sin tcos tdt
sin tx sin tcos tdtx sin2tcos tdt
x2 sin2tcos tdt
sin2ta2
sin t zcos tdt dz
z2dzz2a2
dz a2 dzz2a2
z aarctan za sin t aarctan sin t
a
x2 a2 aarctan x2a2a C
1204* dx
x x21
x 1t
dx dtt2
dtt2
1t
1t21
dt
t2
1t
1t2
t2
dtt
1t2
t2
dtt
1t 1t2
dt
1t2 arcsin t
arcsin 1x C
1205
x21x dx
x sinh tdx cosh t
sinh2t1sinh t cosh t sinh2t1
sinh t cosh t cosh2tsinh t sinh2t1sinh t sinh t dtsinh t
cosh t 2arctanhet cosh arcsinhx 2arctanhearcsinhx C
1206* dx
x2 4x2
x 2sin tdx 2cos tdt
2cos tdt2sin t2 42sin t2
2cos t4sin2t 44sin2t
dt 2cos t4sin2t 44sin2t
dt 2cos t8sin2t 1sin2t
dt 2cos t8sin2t cos2t
2cos t8sin2tcos t
dt 14
dtsin2t
14 tan t
14 tanarcsin
x2 C
dx
x2 4x2
x 1t
dx dtt2
dtt2
1t2
4 1t2
dt
4 1t2
dt1t 4t21
12 t
t2 14dt 12
12 4t2 1
14 4t2 1
14 4 1x
2 1 14x 4 x2 C
1207
1 x2 dx 1x2
1x2dx
1x2
1x2 ax b 1 x2 c
1x2|
1x2
1x2 ax b 1 x2
c
1x2
1x2
1x2 a 1 x2 2xaxb
2 1x2 c
1x2| 1 x2
1 x2 a1 x2 xax b c1 x2 a 2ax2 xb c1 x2 x22a xb a ca 1
2 ,b 0a c 1. . . .c 1
2
I 12 x 1 x
2 12
dx
1x2 1
2 x 1 x2 1
2 arcsinx C
1208 dx
x1x
x sin2tdx 2sin tcos tdt
2sin tcos tsin2t1sin2t
dt 2 sin tcos tsin t 1sin2t
dt 2 sin tcos tsin t cos2t
dt 2 sin tcos tsin tcos t dt 2 dt 2t
2arcsin x C
1209
a2 x2 dx
x a sinh tdx acosh t
a2 a2 sinh2t acosh tdt a2 1 sinh2t cosh tdt
a2 cosh2t cosh tdt a2 cosh2tdt a2 cosh2t12 dt a2
2 cosh2tdt a2
2 dt 1
4 a2 sinh2t 1
2 a2t 1
4 a2 sinh2arcsinh x
a 12 a
2 arcsinh xa
12 x x2 a2 1
2 a2 arcsinh x
a C
1210 x2
x2a2dx
x acosh tdx a sinh tdt
a2 cosh2ta2 cosh2ta2
dt a2 cosh2ta cosh2t1
dt a cosh2tcosh2t1
dt
a 1sinh2tsinh t dt a dtsinh t a sinh tdt acosh t a ln tanh
t2
acosharccosh xa a ln tanh
arccosh xa
2 x a lntanh 12 arccoshxa C
1211 lnxdxdx dv lnx ux v dx
x du
x lnx x dxx x lnx x C
1212
arctanxdxarctanx u dx dvdx1x2
du x v
xarctanx x1x2
dx
1 x2 t2xdx dt
xarctanx dt2t xarctanx 1
2 ln t xarctanx 12 ln1 x
2 C
1213 arcsinxdxarcsinx u dx dvdx
1x2 du x v
xarcsinx x
1x2dx
1 x2 t2xdx dt
xarcsinx dt2
t xarcsinx 1
2 dtt xarcsinx t xarcsinx 1 x2 C
1214
x sinxdx
x u sinxdx dvdx du cosx v
xcosx cosxdx xcosx cosxdx xcosx sinx C1215 xcos3xdx
3x t3dx dt
19 tcos tdtt u cos tdt dvdt du sin t v
19 t sin t sin tdt
19 t sin t
19 cos t
19 3x sin3x
19 cos3x
13 x sin3x
19 cos3x C
1216 xex dx
x u exdx dv
dx du ex v
xex exdx xex ex C1217 x2xdx
x u 2xdx dvdx du 1
ln2 2x v
1ln2 x2
x 1ln2 2
xdx 1ln2 x2
x 1ln2 2xdx
1ln2 x2
x 1ln222x C
1218
x2e3xdx3x t3dx dt
t29 etdt3 1
27 t2etdtetdt dv t2 u
et v 2tdt du
I 127 t2et 2 ettdt
I1 ettdt
t u etdt dv
dt du et v
tet etdt tet et
I 127 t
2et 2I1 127 t
2et 2tet et 127 t
2et 227 te
t 227 e
t
127 3x
2e3x 227 3xe
3x 227 e
3x 13 x
2e3x 29 xe
3x 227 e
3x C
1219
x2 2x 5exdx x2exdx 2 xexdx 5 exdxx2 u exdx dv
2xdx du ex v
I1 exx2 ex2xdx exx2 2 xexdxx u exdx dv
dx du ex v
I12 exx exdx exx exdx exx ex
I1 exx2 2exx ex exx2 2exx 2ex
I2 2 xexdx 2exx ex 2exx 2ex
I3 5exI I1 I2 I3 exx2 2exx 2ex 2exx 2ex 5exI exx2 5ex C
1220* x3e x3
x3 t
dx3 dt
27t3et3dt 81 t3etdt 81I1I1 t3etdtetdt dv t3 u
et v 3t2dt du
t3et 3 ett2dt t3et 3I2I2 ett2dtetdt dv t2 u
et v 2tdt du
I2 t2et 2 ettdt t2et 2I3I3 ettdtetdt dv t u
et v dt du
tet etdt tet et
I2 t2et 2I3 t2et 2tet et t2et 2tet 2etI1 t3et 3I2 t3et 3t2et 2tet 2et t3et 3t2et 6tet 6et
I 81I1 81t3et 3t2et 6tet 6et 81 x3
3e
x3 243 x3
2e
x3 486 x3 e
x3 486e x3
3x3e 13 x 27x2e 13 x 162xe 13 x 486e 13 x C1221 x sinxcosxdx 1
2 x sin2xdx2x t2dx dt
12
t2 sin t
dt2 1
8 t sin tdtu t dv sin tdtdu dt v cos t
18 tcos t cos tdt 1
8 tcos t cos tdt 18 tcos t
18 sin t
18 2xcos2x
18 sin2x
14 xcos2x
18 sin2x C
1222*x2 5x 6cos2xdx
2x t2dx dt
t2
4 52 t 6 cos t dt2 1
8 t2 cos tdt 54 tcos tdt 3 cos tdt
3sin t 18 t2 cos tdt
54 tcos tdt
t2 u cos tdt dv2tdt du sin t v
t u cos tdt dvdt du sin t v
Ia t2 sin t 2 t sin tdtIb t sin t sin tdt t sin t cos tIaa t sin tdt
t u sin tdt dvdt du cos t v
tcos t cos tdt tcos t sin tIa t2 sin t 2tcos t sin t t2 sin t 2tcos t 2sin tI 3sin t 1
8 Ia 54 Ib 3sin t
18 t
2 sin t 2tcos t 2sin t 54 t sin t cos t
114 sin t
18 t2 sin t 1
4 tcos t 54 t sin t
54 cos t
114 sin2x
18 2x
2 sin2x 14 2xcos2x
54 2x sin2x
54 cos2x
114 sin2x
12 x
2 sin2x 12 xcos2x
52 x sin2x
54 cos2x C
1223 x2 lnxdx
lnx u x2dx dvdxx du x3
3 v
x3
3 lnx x3
3dxx x3
3 lnx 13 x2dx
13 x
3 lnx 19 x
3 C
1224 ln2xdx
ln2x u dx dv2 lnxx dx du x v
x ln2x x 2 lnxx dx x ln2x 2 lnxdxI1 lnxdx
lnx u dx dvdxx du x v
I1 x lnx x dxx x lnx xI x ln2x 2x lnx x x ln2x 2x lnx 2x C
1225
lnxx3dx
lnx u x3dx dvdxx du x2
2 v
lnx x22 x2
2dxx 12
lnxx2
12
dxx3
12lnxx2 1
4x2 C
1226
lnxxdx
lnx u dxx dv
dxx du 2 x v
2 x lnx 2 x dxx 2 x lnx 2 dx
x 2 x lnx 4 x C
1227
xarctanxdxarctanx u xdx dvdx1x2
du x2
2 v
x2
2 arctanx x2
2dx1x2
x2
2 arctanx 12
x2
1x2dx x2
2 arctanx 12
x2111x2
dx
x2
2 arctanx 12 dx
12
dx1x2
12 x
2 arctanx 12 x
12 arctanx C
1228 xarcsinxdx
xdx dv arcsinx u12 x
2 v dx
1x2 du
12 x
2 arcsinx 12 x2dx
1x2
12 x
2 arcsinx 12
1x211x2
dx 12 x
2 arcsinx 12 1 x2 dx 1
2 dx
1x2
12 x
2 arcsinx 14 x 1 x2 1
4 arcsinx C
1229
ln x 1 x2 dx
dx dv ln x 1 x2 u
x v dx
1x2 du
x ln x 1 x2 x dx
1x2 x ln x 1 x2 x
1x2dx
x ln x 1 x2 x
1x2dx
1 x2 t2xdx dt
x ln x 1 x2 dt2
t
x ln x 1 x2 t x ln x 1 x2 1 x2 C
1230 xdxsin2x
x u dxsin2x
dv
dx du cotx v
xcotx cotxdx xcotx cotxdx xcotx lnsinx C
1231
I xcosxsin2x
cosxsin2x
dx dv x u
1sinx v dx du
I1 cosxsin2x
sinx tcosxdx dt
dtt2 1
t 1sinx
I x 1sinx 1
sinx dx xsinx dx
sinx xsinx ln tan
x2 C
1232
ex sinxdxexdx dv sinx uex v cosxdx du
I ex sinx ex cosxdxI1 ex cosxdxexdx dv cosx uex v sinxdx du
ex cosx ex sinxdxI1 ex cosx ex sinxdxI ex sinx ex cosx ex sinxdx ex sinx ex cosx I2I ex sinx ex cosxI ex sinxex cosx
2 C ex sinxdx
ex u sinxdx dvexdx du cosx v
ex cosx cosxexdx ex cosx cosxexdx
ex u cosxdx dvexdx du sinx v
I ex cosx ex sinx sinxexdxI ex cosx ex sinx I2I ex cosx ex sinxI ex cosxex sinx
2 C
1233 3x cosxdx3xdx dv cosx u3xln3 v sinx du
3xln3 cosx sinx
3xln3 dx
I 3xln3 cosx
1ln3 3x sinxdx
3xdx dv sinx u3xln3 v cosx du
I 3xln3 cosx
1ln3 sinx 3x
ln3 3xln3 cosx 3x
ln3 cosx 3xln23
sinx 1ln23 3x cosx
I 3xln3 cosx
3xln23
sinx 1ln23I
I 1 1ln23
3xln3 cosx
3xln23
sinx
I 3xln3 cosx
3x
ln23sinx
ln231ln23
3xln3 cosx
3x
ln23sinx
ln231ln23 3x cosx ln3sinx
ln231 C
1234 eax sinbxdx
bx t
bdx dt
e atb sin t dt
b 1
b e at
b sin tdt
eatb u sin tdt dv
abea
tb dt du cos t v
1be
atb cos t a
b ea tb cos tdt
eatb u cos tdt dv
abea
tb dt du sin t v
I 1beatb cos t a
b2eatb sin t a 1
b ea tb sin tdt
I 1beatb cos t a
b2eatb sin t aI
I 1beatb cos t a
b2eatb sin t a2
b2I
I 1 a2
b2 1
beatb cos t a
b2eatb sin t
I 1beatb cos t a
b2eatb sin t
a2b2
b2
eatb bcos ta sin t
a2b2 ea
bxbbcosbxa sinbx
a2b2
I bcosbxa sinbxa2b2
eax C
1235 sinlnxdx
lnx tdxx dt
x sin tdt et sin tdtetdt dv sin t uet v cos tdt du
I et sin t et cos tdtI1 et cos tdtetdt dv cos t uet v sin tdt du
I1 et cos t et sin tdt et cos t et sin tdtI et sin t et cos t et sin tdt et sin t et cos t I2I et sin t et cos tI et sin tet cos t
2 elnx sin lnxelnx cos lnx2 1
2 x sinlnx 12 xcoslnx C
1236 x3ex2dx
x2 t
2xdx dt
x3et dt2x 12 x2etdt
12 tetdt
t u etdt dv
dt du et v
12 tet etdt 1
2 tet 1
2 et
12 x
2ex2 12 e
x2 12 x
2ex2 1
2 ex2 C
1237 e x dx
x t|2
x t2
dx 2tdt
2 ettdtt u etdt dv
dt du et v
2 tet etdt 2tet 2et 2 x e x 2e x C
1238x2 2x 3 lnxdx
x2 2x 3dx dv lnx ux3
3 x2 3x v dx
x du
x3
3 x2 3x lnx x3
3 x2 3x dxx x3
3 x2 3x lnx x2
3 x 3 dx
x3
3 x2 3x lnx x3
9 x2
2 3x C
1239 x ln 1x
1x dx
ln 1x1x u dv xdx
2x21
dx du v x2
2
x2
2 ln1x1x
x2
22x21
x2
2 ln1x1x
x2
x21 x2
2 ln1x1x
x211x21
x2
2 ln1x1x dx
dxx21
12 x
2 ln 1x1x x arctanhx C
1240 ln2x
x2dx
ln2x u dv dxx2
2 lnxx dx du v 1x
1x ln2x 1x 2 lnxx dx 1x ln2x 2 lnxx2 dx
lnx u dv dxx2
dxx du v 1x
1x ln2x 2 1x lnx 1x dxx 1x ln2x 2 1x lnx dxx2 1
x ln2x 2x lnx 2
x C
1241 lnlnxx dx
lnlnx u dxx dv
dxx lnx du lnx v
lnx lnlnx lnx dxx lnx lnx lnlnx dxx lnx lnlnx lnx C
1242
x2 arctan3xdx3x t3dx dt
t29 arctan tdt3 1
27 t2 arctan tdtarctan t u t2dt dvdt1t2
du t3
3 v
127
t3
3 arctan t t3
3dt1t2
127
t3
3 arctan t 13
t3
1t2dt 1
81 t3 arctan t t3
1t2dt
1 t2 z2tdt dz
181 t3 arctan t t3z dz
2t
181 t3 arctan t 1
2 t2z dz 1
81 t3 arctan t 12
z1z dz
181 t3 arctan t 1
2 dz 12
dzz 1
81 t3 arctan t 1
2 z 12 ln z
181 t
3 arctan t 12 1 t
2 12 ln1 t
2 1
81 3x3 arctan3x 12 1 3x2 1
2 ln 1 3x2
181 27x
3 arctan3x 12
92 x
2 12 ln1 9x
2 1
3 x3 arctan3x 1
162 118 x
2 1162 ln1 9x
2 C
1243 xarctan2xdx
arctan2x u xdx dv2arctanx1x2
dx du x2
2 v
x2
2 arctan2x x22
2arctanx1x2
x2
2 arctan2x x2 arctanx
1x2
arctanx u x2
1x2dx dv
dx1x2
du x2111x2
dx x arctanx v
x2
2 arctan2x x arctanxarctanx x arctanx dx
1x2
x2
2 arctan2x xarctanx arctan2x x
1x2dx arctanx
1x2dx
x2
2 arctan2x xarctanx arctan2x I1 I2
I1 x1x2
dx
1 x2 t2xdx dt
12
dtt 1
2 ln t 12 ln1 x
2
I2 arctanx1x2dx
arctanx tdx1x2
dt
tdt 12 t2 1
2 arctan2x
x2
2 arctan2x xarctanx arctan2x 1
2 ln1 x2 1
2 arctan2x
12 x
2 arctan2x xarctanx 12 arctan
2x 12 ln1 x
2 C
1244I arcsinx2dx
dv dx arcsin2x uv x 2 arcsinx
1x2 dx du
I xarcsin2x x2 arcsinx1x2
dx
Iv x arcsinx1x2
arcsinx u x
1x2 dx dv
11x2
dx du 1 x2 v
Iv uv vdu 1 x2 arcsinx 1 x2 11x2
dx 1 x2 arcsinx x
Iv x
1x2
1 x2 t2xdx dt
12dtt 12 t
12 dt 12 2t12 t 1 x2
I xarcsin2x 2 1 x2 arcsinx x xarcsin2x 2 1 x2 arcsinx 2x C1245 arcsinx
x2dx
arcsinx u dxx2
dvdx
1x2 du 1x v
1 x2 t
1 x2 t2
xdx tdt
1x arcsinx 1x dx
1x2 1x arcsinx dx
x 1x2
I1 tdtxxt
dtxx dt
x2 dt
1t2 arctanh t arctanh 1 x2
I 1x arcsinx I1 1x arcsinx arctanh 1 x2 C
1246
arcsin x
1xdx
x t
x t2
dx 2tdt
I 2 tarcsin t1t2
dt
arcsin t1t2
dt dv t u
12 arcsin
2t v dt du
I 2 t 12 arcsin2t 12 arcsin
2tdt tarcsin2t arcsin2tdtI1 arcsin2tdt
arcsin2t u dt dv2arcsin t1t2
dt du t v
tarcsin2t 2 tarcsin t1t2
dt
arcsin t u tdt
1t2 dv
dt
1t2 du v 1 t2
I2 1 t2 arcsin t 1 t2 dt
1t2 arcsin t 1 t2 dt
I2 arcsin t 1 t2 tI1 tarcsin2t 2 arcsin t 1 t2 t tarcsin2t 2arcsin t 1 t2 2t
I tarcsin2t I1 tarcsin2t tarcsin2t 2arcsin t 1 t2 2t
I 2arcsin t 1 t2 2t 2arcsin x 1 x 2 2 x
I 2 1 x arcsin x 2 x C
1247 x tan22xdx
2x t2dx dt
14 t tan2tdttan2t dv u t
tan t t v du dt
I 14 ttan t t tan t tdt 1
4 t tan t t2 tan tdt tdtI 1
4 t tan t 12 t2 lncos t 1
4 2x tan2x 2x2 lncos2x C
1248 sin2x
ex dx
dxex dv sin2x u
ex v 2sinxcosxdx sin2xdx du
I ex sin2x ex sin2xdx ex sin2x I1I1 ex sin2xdx
sin2x u dxex dv
2cos2xdx du ex v
ex sin2x 2 ex cos2xdxI1 ex sin2x 2I2
cos2x u dxex dv
2sin2xdx du ex v
I2 ex cos2x 2 ex sin2xdx ex cos2x 2I1I1 ex sin2x 2ex cos2x 2I1
ex sin2x 2ex cos2x 4I1 ex sin2x2ex cos2x5
I ex sin2x I1 ex sin2x ex sin2x2ex cos2x5 C
1249 cos2lnxdx
cos2lnx u dx dv
2cos lnx sin lnx dxx du x v
I xcos2lnx x2cos lnx sin lnx dxx xcos2lnx x2cos lnx sin lnx dxxI xcos2lnx sin2 lnxdx xcos2lnx I1
sin2 lnx u dx dv2x cos2 lnxdx du x v
I1 sin2 lnxdx x sin2 lnx x 2x cos2 lnxdx
I1 x sin2 lnx 2 cos2 lnxdx x sin2 lnx 2I2I2 cos2 lnxdx
cos2 lnx u dv dx
2x sin2 lnxdx du v x
I2 xcos2 lnx x 2x sin2 lnxdx xcos2 lnx 2 sin2 lnxdxI2 xcos2 lnx 2I1I1 x sin2 lnx 2xcos2 lnx 2I1 x sin2 lnx 2xcos2 lnx 4I1I1
x sin2 lnx2xcos2 lnx5
I xcos2lnx x sin2 lnx2xcos2 lnx5 C
1250** x2
x212dx
x u xdx
x212 dv
dx du 121x2
v
121x2
x 121x2
dx
x21x2
12
11x2
dx x22x2
12 arctanx C
1251* dx
x2a2 2 1
a2 x2a2x2
x2a2 2dx 1
a2 dxx2a2
1a2 x2
x2a2 2dx
I 1a3arctan x
a 1a2 x2
x2a2 2dx
x at
dx adt
I 1a3arctan x
a 1a2 a2t2
a2t2a2 2adt 1
a3arctan x
a a t2
a4t212dt
I 1a3arctan x
a 1a3 t2
t212dt
t
t212dt dv t u
121t2
v dt du
I 1a3arctan x
a 1a3 121t2
t 121t2
dt 1a3arctan x
a 12a3
t1t2
dt1t2
I 1a3arctan x
a 12a3
t1t2
arctan t
I 1a3arctan x
a 12a3
xa
1 xa 2 arctan xa C
1252**
a2 x2 dx
x a sin tdx acos tdt
a2 a2 sin2t acos tdt
a2 1 sin2t cos tdt a2 cos2tdt a22 1 cos2tdt a22 t
a2
2 cos2tdt 12 a
2t 14 a
2 sin2t 1
2 a2 arcsin x
a 14 a
2 sin2arcsin xa C
1253*
A x2
x A sinh tdx A cosh tdt
A A sinh2t A cosh tdt A 1 sinh2t cosh tdt A cosh2tdt A2 cosh2t 1
A4 sinh2t
A2 t
A4 sinh2 arcsin
xA
A2 arcsin x
A C
A x2 Ax2
Ax2
Ax2
Ax2 ax b A x2 c dx
Ax2|
Ax2
Ax2 a A x2 2xaxb
2 Ax2 c
Ax2| A x2
A x2 aA x2 xax b cA x2 aA 2ax2 xb caA c A2a 1
c A aA A1 a A2
b 0,a 12 ,c
A2
I 12 x A x2 A2
dx
Ax2
I 12 x A x
2 12 A ln x A x2 C
1254*
x2
9x2dx 9x29
9x2dx 9 x2 9 dx
9x2 9 x2 9arcsin x
3
9x2
9x2 ax b 9 x2 c
9x2dx|
9x2
9x2 a 9 x2 axbx
9x2 c
9x2| 9 x2
9 x2 a9 x2 ax bx c9 x2 9a 2ax2 xb c9 x2 x22a xb c 9ac 9a 9. . .c 9
2 9
a 12 , b 0 , c 9
2
I1 x2 9 x2 9
2 dx
9x2 x
2 9 x2 92 arcsin
13 x
I 9arcsin x3 I1 9arcsin
x3
x2 9 x2 9
2 arcsinx3
I 92 arcsin
x3
x2 9 x2 C
1255 dxx22x5
dxx22x14
dxx124
12 arctan
x12 C
1256 dxx22x
dxx22x11
dxx121
arctanhx 1 C
1257 dx3x2x1
13
dxx2 13 x
13 1
3 dx
x 162 1
36 13
13
dx
x 162 1136
211arctan 6x1
11 C
1258
xdxx27x13
xdx
x 722 494 13
x 72x 72
2 34dx 7
2 dx
x 722 34
x 72 t
dx dt
tt2 34
dt 72
dx
x 722 34
12 ln4t
2 3 73 3 arctan 1
3 2x 7 3
12 ln 4x
72
2 3 73arctan 2x7
3
ln2 12 lnx
2 7x 13 73arctan 2x7
3 C
1259 3x2x24x3
dx 3x2x2243
dx 3x2x221
dx 3 xdxx221
2 dxx221
3 x22x221
dx 2 dxx221
3 x2x221
dx 6 dxx221
2 dxx221
3 x2x221
dx 4 dxx221
x 2 tdx dt
t2 1 z2tdt dz
3 tt21dt 4 dt
t21 3 t
t21dt 4arctanh t 3
2 dzz 4arctanh t
32 ln z 4arctanh t
32 lnt
2 1 4arctanh t 32 ln x 22 1 4arctanhx 2 3
2 lnx 1 32 lnx 3 4arctanhx 2
1260
x12
x23x4dx x22x1
x23x4dx x23x42x13x4
x23x4dx x23x4
x23x4dx 5x3
x23x4
x 52
2x 65x23x4
x 52
2x 65 33
x 322 94 4
x 52
2x3 95x 32
2 74
x 52
2x3x23x4
92
dx
x 322 74
x2 3x 4 t x 32 z
2x 3dx dt dx dz
x 52
dtt
92
dzz2 74
x 52 ln t
97 7 arctan 2
7 z 7
x 52 lnx
2 3x 4 97arctan 2x3
7 C
1261
x2
x26x10dx x26x106x10
x26x10dx dx 6x10
x32910dx x 3 2x6
103 6
x321dx
x 3 2x6x26x10
dx 8 dxx321
x2 6x 10 t x 3 z2x 6dx dt dx dz
x 3 dtt 8 dzz21
x 3 ln t 8arctan z x 3 lnx2 6x 10 8arctanx 3 C
1262 dx
33x2x2 1
2 dx
32
32 xx
2 1
2 dx
3316 x
34
2
x 34 t
dx dt
12 dt
3316 t
2 1
2arcsin 4t
33 1
2arcsin 4x3
33 C
1263 dx
xx2 dx
14 x
12
2
x 12 t
dx dt
dt14 t
2 arcsin2t arcsin2x 1 C
1264 dx
x2pxq dx
x p22 p
24 q
dx
x p22 p
24 q
dx
x p22 p
24q4
x p2 t
dx dtp24q4 z
dt
t2z ln t t2 z ln x p
2 x p2
2 p24q4
ln 12 p x x2 xp q C
1265 3x6
x24x5dx 3
2 2x4x24x5
dx
x2 4x 5 t2x 4dx dt
32
dtt 3 t 3 x2 4x 5 C
1266
2x81xx2
dx 2x8
1 x 122 14
dx 2 x454 x
12
2dx 2 x 12 4
12
54 x
12
2dx
2 x 1254 x
12
2dx 9 dx
54 x
12
2
54 x
12
2 t x 12 z
2x 12 dx dt dx dz
dtt 9 dz
54 z
2 2 t 9arcsin 2
5 5 z 2 1 x x2 9arcsin 2x15 C
1267 x
5x22x1dx 1
10 10x22x2 25 x
15
dx 110
10x25x22x1
dx 15 5 dx
x 152 1
25 15
5x2 2x 1 t x 15 z
10x 2dx dt dx dz
110
dtt 1
5 5 dx
x 152 4
25
110 2t
12 1
5 5 dz
z2 425
t5 5
25 arcsinh52 z
5x22x15 5
25 arcsinh5x12 C
1268 dx
x 1x2
1 x2 t
1 x2 t2
xdx tdt
tdtxxt tdt
x2t dt
1t2 arctanh t arctanh 1 x2 C
1269 dx
x x2x1
x 1t
dx dtt2
dtt2
1t
1t2 1t 1
dtt1t2 1t 1
dtt1tt2
t2
dtt
1t 1tt2
dt
1tt2 dt
1 t 122 14
t 12 z
dt dz
dz54 z
2 arcsin 2
5z arcsin 2
5t 1
2
arcsin 25 1x 1
2 arcsin25 2x2x arcsin
2xx 5
C
1270 dx
x1 x22
x 1 tdx dt
dt
t t122 dt
t t22t1
t 1z
dt dzz2
dzz2
1z
1z2 2z 1
dzz12zz2
z2
dz
12zz2 dz
2z12 arcsin z1
2
arcsin1t 1
2 arcsin
1x1 1
2 arcsin
1x1x1
2 arcsin
2xx1
2
arcsin 2x2 x1
C
1271 dx
x1 x22x dx
x1 x121
x 1 tdx dt
dt
t t21
t 1z
dt dzz2
dz
z2
1z 1z
21
dzz
1z 21
dzz
1z2
z2
dz
1z2 arcsin z
arcsin 1t arcsin 1
x1 C
1272
x2 2x 5 dx x2 2x 1 4 dx x 12 4 dx
x 1 tdx dt
t2 4 dt
t 2sinh zdt 2cosh zdz
4sinh2z 4 2cosh zdz 4 sinh2z 1 cosh zdz 4 cosh2zdz 4 12 1 cosh2zdz 2 1 cosh2zdz 2 dz 2 cosh2zdz
2z u2dz du
2z coshudu 2z sinhu 2z sinh2z 2arcsinh t
2 sinh2arcsinht2 2arcsinh
x12 sinh2arcsinh x1
2 C
1273
x x2 dx 14 x
12
2dx
x 12 t
dx dt
14 t
2 dt 14 t
2
14 t
2dt
14 t
2
14 t
2dt at b 1
4 t2 c
14 t
2dt|
14 t
2
14 t
2 a 1
4 t2 2tatb
2 14 t
2 c
14 t
2| 1
4 t2
14 t
2 a 14 t2 tat b c
14 t
2 t22a tb c 14 a
a 12 ,b 0
.c 14 a
14 . . . .c
18 1
4c 1
8
I t2
14 t
2 18
dt14 t
2 1
4 t 1 4t2 1
8 arcsin2t
14 x
12 1 4x 1
2 2 1
8 arcsin2x 12
14 2x 1 x x2 1
8 arcsin2x 1 C
1274
2 x x2 dx 2 14
14 x x
2 dx 94 x
12
2
x 12 t
dx dt
94 t
2 dt 14 t 9 4t
2 98 arcsin
23 t
14 x
12 9 4x 1
2 2 9
8 arcsin23 x
12
12 x 2 x
2 x 14 2 x2 x 9
8 arcsin23 x
13 C
1275 xdxx44x23
xdxx44x241
xdx
x2221
x2 2 t2xdx dt
12
dtt21
12 arctanh t
12 arctanhx
2 2 C
1276 cosxsin2x6sinx12
dx cosxsinx323
dx
sinx 3 tcosxdx dt
dtt23
13arctan t
3 1
3arctan sinx3
3 C
1277 ex
1exe2xdx
ex t
exdx dt
dt
1tt2 dt
1 t 122 14
dt
34 t 12
2 arcsinh 2t1
3 arcsinh 2ex1
3 C
1278 sinxdx
cos2x4cosx1 sinx
cosx223dx
cosx 2 t sinxdx dt
dt
t23 ln t t2 3 ln cosx 2 cos2x 4cosx 1 C
1279I lnxdx
x 14 lnxln2x lnxdx
x lnx2241 lnxdx
x 5lnx22
lnx udxx du
I udu
5u22
u 2 zdu dz
z25z2
dz zdz
5z2 2 dz
5z2
I1 zdz
5z2
5 z2 t2zdz dt
dt2
t t 5 z2
I1 5 u 22 1 u2 4u 1 ln2x 4 lnxI2 dz
5z2 arcsin z
5 arcsin u2
5 arcsin lnx2
5
I I1 2I2 1 ln2x 4 lnx 2arcsin lnx25
C
1280I dx
xaxb
1xaxb
Axa
Bxb
1 Ax b Bx a1 Ax Ab Bx Ba1 xA B Ab BaA B 0Ab Ba 1. . . .Ba Bb 1Ba b 1B 1
ab ,A 1ba
I 1ba
xa
1ab
xb 1
ba dxxa
1ab
dxxb
I 1ba lnx a
1ab lnx b C
1281I x25x9
x25x6dx dx 3 dx
x25x6 x 3 dx
x2x31
x2x3 ax2
bx3 |x 2x 3
1 ax 3 bx 2 ax 3a bx 2b xa b 3a 2ba b 0. . .a b3a 2b 13b 2b 1b 1,a 1I x 1x2
1x3 x 3 lnx 2 3 lnx 3 C
1282 dxx1x2x3 1
12 dxx1
13
dxx2
14
dxx3
112 lnx 1
13 lnx 2
14 lnx 3
1x1x2x3 a
x1 b
x2 c
x3 |x 1x 2x 31 ax 2x 3 bx 1x 3 cx 1x 21 ax2 5ax 6a bx2 2bx 3b cx2 cx 2c1 x2a b c x5a 2b c 6a 3b 2ca b c 05a 2b c 06a 3b 2c 1b 13 ,a
112 ,c
14
1283 2x241x91x1x3x4 dx2x241x91
x1x3x4 ax1
bx3
cx4 |x 1x 3x 4
2x2 41x 91 ax 3x 4 bx 1x 4 cx 1x 32x2 41x 91 ax2 ax 12a bx2 5bx 4b cx2 2cx 3c2x2 41x 91 x2a b c xa 5b 2c 12a 4b 3c2 a b c
41 a 5b 2c91 12a 4b 3c
43 4b 3c c 433 4
3 b
12a 4b 3 433 4
3 b 12a 43 914 b 43
3 43 b
553 7
3 b 22 4 7 c 3 cb 7,c 5,a 4I 4 dx
x1 7 dxx3 5
dxx4 4 lnx 1 7 lnx 3 5 lnx 4 C
1284
5x32x35x24x
dx 5x32xx25x4
dx 5x32xx1x4 dx 5
x3 25xx1x4 dx
5 x35x24x 25 5x
24xxx1x4 dx 5 dx 25x220x2
xx1x425x220x2xx1x4 a
x bx1
cx4 |xx 1x 4
25x2 20x 2 ax2 5x 4 bxx 4 cxx 125x2 20x 2 ax2 5ax 4a bx2 4bx cx2 cx25x2 20x 2 x2a b c x5a 4b c 4a25 a b c20 5a 4b c2 4aa 1
2 4a 3b 4 12 3b 2 3b 5b 7312
73 c 25
c 1616
I1 12
dxx 7
3 dxx1
1616
dxx4 1
2 lnx 73 lnx 1
1616 lnx 4
I 5x I1 5x 12 lnx
73 lnx 1
1616 lnx 4 C
1285 dxxx12
x 1t
dx dtt2
dtt2
1t
1t 1
2 dt
t 1tt 2 t
1t2dt t11
1t2dt
dt1t
dt1t2
ln1 t 11t ln
x1x x
x1 C
1286 x314x3x
dx 14
4x344x3x
dx 14
4x3xx44x3x
dx 14 dx
14
x44x3x
dx
x4
14
x4x4x21
dx x4
14
x4x2x12x1 dx
x4x2x12x1 a
x b2x1
c2x1 |x2x 12x 1
x 4 a2x 12x 1 bx2x 1 cx2x 1x 4 4ax2 a 2bx2 bx 2cx2 cxx 4 x24a 2b 2c xb c aa 44a 2b 2c 0b c 116 21 c 2c 0c 92 b 1 c 1 9
2 72
I 4x dx 72
2x1 dx 92
2x1 dx 4 lnx 74 ln2x 1
94 ln2x 1 C
1287I x46x312x26
x36x212x8dx x46x312x26
x23dx
x4 6x3 12x2 6 : x3 6x2 12x 8 x 68xx36x212x8
x4 6x3 12x2 8x6 8xI xdx 2 4x3
x23dx x2
2 2 4x838x23
dx x2
2 8 x2x23
dx 22 dxx23
x2
2 8 dxx22
22 dxx23
x2
2 8x2
11x22
C
1288I 5x26x9
x32x12dx
I 5x26x9x22x32
dx axbx22x3
cxdx22x3
dx|
5x26x9x22x32
axbx22x3
cxd
x22x3
5x26x9x22x32
ax22x32x2axb
x22x32 cxd
x22x3|x2 2x 32
5x2 6x 9 ax2 2x 3 2x 2ax b cx dx2 2x 35x2 6x 9 ax2 3a 2xb 2b cx3 2cx2 3cx dx2 2dx 3d5x2 6x 9 x3c x2a 2c d x2b 3c 2d 3a 2b 3dc 0a 2c d 5 d a. . .d 5 a2b 3c 2d 6 2d 2b9 3a 2b 3d
15 2d 2b 3a 2b 3d 5d 3a55 a 3a 8a 25 15a 5,d 06 2b. . .b 3I 5x26x9
x22x32dx 5x3
x22x3 0
x22x3dx 5x3
x22x3 C
1289 x28x7x23x102
dx x28x7x22x52
dx
x28x7x22x52
ax2
bx22
cx5
dx52
|x 22x 52
x2 8x 7 ax3 8ax2 5ax 50a bx2 10bx 25b cx3 cx2 16cx 20c dx2x2 8x 7 x3a c x28a b c d x5a 10b 16c 4d 50a 25b 20ca c 08a b c d 15a 10b 16c 4d 850a 25b 20c 4d 7a 30
343 ,c 30343 ,b
2749 ,d
849
I 30343 dx
x2 2749 dx
x22
30343 dx
x5 849 dx
x52 27
49x2 30343 lnx 2
849x5
30343 ln
1290 2x3x23x23
dx
x2 3x 2 t2x 3dx dt
dtt3 1
2t2 1
2x23x22 C
1291 x3x1xx21
dx x3x1x3x
dx dx dxxx21
1xx21
ax bxc
x21|xx2 1
1 ax2 a bx2 cxa b 0c 0a 1b 1I1 dxx x
x21dx lnx 1
2 lnx2 1
I x I1 x lnx 12 lnx
2 1 C
1292 x4
x41dx x411
x41dx dx x dx
x41 x I1
1x21x21
axbx21
cxdx21
|x2 1x2 1
1 ax3 ax bx2 b cx3 cx dx2 d1 x3a c x2b d xa c b da c 0 a cb d 0 b da c 0 2c 0 c 0a 0b d 1 2d 1 d 12 b
12
I1 12
x21dx
12
x21dx 1
2 arctanhx 12 arctanx
I x 12 arctanhx
12 arctanx C
1293 dxx24x3x24x5
dxx3x1x24x5
x12 4 1612
2x1 42
2 3x2 42
2 11
x3x1x24x5 a
x3 bx1
cxdx24x5
|x 3x 1x2 4x 5
1 ax3 3ax2 ax 5a bx3 bx2 7bx 15b cx3 4cx2 3cx dx2 4dx 3d1 x3a b c x23a b 4c d xa 7b 3c 4d 5a 15b 3d
1 1 1 0 03 1 4 1 01 7 3 4 05 15 0 3 1
3 1 5
1 1 1 0 00 2 7 1 00 8 2 4 00 10 5 3 1
: 4: 5
1 1 1 0 00 2 7 1 00 2 1
2 1 0
0 2 1 35 15
1 1 1 0 00 2 7 1 00 2 1
2 1 0
0 0 12 1 35 15
1 1 1 0 00 2 7 1 00 2 1
2 1 0
0 0 12 85 15
1 1 1 0 00 2 7 1 00 2 12 1 0
0 0 12 85 15
1 1 1 0 00 2 7 1 00 0 7 1
2 2 0
0 0 12 85 15
1 1 1 0 00 2 7 1 00 0 152 2 0
0 0 12 85 15 15
1 1 1 0 00 2 7 1 00 0 152 2 0
0 0 152
8155 3
1 1 1 0 00 2 7 1 00 0 152 2 0
0 0 152 24 3
1 1 1 0 00 2 7 1 00 0 152 2 0
0 0 0 26 3d 3
26 ,b 120 ,a
152 ,c
265
I 152
dxx3
120
dxx1
265 x
326
x24x5dx
I 152 lnx 3
120 lnx 1
165
2x 326 65
x24x5dx
I 152 lnx 3
120 lnx 1
165
2x 152x24x5
dx
I1 165
2x4 72x24x5
165
2x4x24x5
72
165
dxx24x5
x2 4x 5 t2x 4dx dt
x 2 zdx dz
I1 165
dtt
72
165
dxx221
165 ln t
7130
dzz21
I1 165 ln t
7130 arctan z
165 lnx
2 4x 5 7130 arctanx 2
I 152 lnx 3
120 lnx 1
165 lnx
2 4x 5 7130 arctanx 2 C
1294 dxx31
dxx1x2x1
1x1x2x1
ax1
bxcx2x1
|x 1x2 x 1
1 ax2 bx2 ax bx cx c a1 x2a b xa b c c aa b 0a b c 0c a 1
1 1 0 01 1 1 01 0 1 1
1 1 0 00 2 1 00 1 2 1 2
1 1 0 00 2 1 00 2 4 2
1 1 0 00 2 1 00 0 3 2
b 13 ,c 23 ,a
13
I 13x1 dx
13 x23
x2x1dx 1
3 lnx 1 13
x2x2x1
I 13 lnx 1
16
2x4x2x1
I 13 lnx 1
16
2x1x2x1
12
dxx2x1
I 13 lnx 1
16 lnx
2 x 1 12
dx
x 122 14 1
I 13 lnx 1
16 lnx
2 x 1 12
dx
x 122 34
I 13 lnx 1
16 lnx
2 x 1 33 arctan
2x13 C
1295 dxx41
dx
x2122x211 dx
x2122x2 dx
x2 2 x1 x2 2 x11
x2 2 x1 x2 2 x1 axb
x2 2 x1 cxd
x2 2 x1
1 ax b x2 2 x 1 cx d x2 2 x 11 ax3 ax a 2 x2 bx2 b b 2 x cx3 cx c 2 x2 dx2 d d 2 x
1 x3a c x2 a 2 b c 2 d x a b 2 c d 2 b da c 0a 2 b c 2 d 0a b 2 c d 2 0b d 1
1 0 1 0 02 1 2 1 01 2 1 2 00 1 0 1 1
: 21
~
1 0 1 0 0
1 22 1 2
2 0
1 2 1 2 00 1 0 1 1
~
1 0 1 0 0
0 22 2 2
2 0
0 2 0 2 00 1 0 1 1
22
: 21
~
1 0 1 0 00 1 4
21 0
0 1 0 1 00 1 0 1 1
~
1 0 1 0 00 1 4
21 0
0 0 422 0
0 0 420 1
~
1 0 1 0 00 1 4
21 0
0 0 422 0
0 0 0 2 1
d 12 ,c
24 ,b
12 ,a
24
I 24 x
12
x2 2 x1
24 x
12
x2 2 x1 1
4 2 x2x2 2 x1
14
2 x2x2 2 x1
I 24
x 2x2 2 x1
24
x 2x2 2 x1
28
2x2 2x2 2 x1
28
2x2 2x2 2 x1
I 28
2x 2 2x2 2 x1
28
2x 2 2x2 2 x1
x2 2 x 1 t2x 2 dx dt
x2 2 x 1 z2x 2 dx dz
I 28
dtt
28
dzz 2
8 2
x2 2 x1 2
8 2
x2 2 x1
I 18 2 ln t 18 2 ln z 2
8 dx
x2 2 x1 2
8 dx
x2 2 x1
I 18 2 ln t 18 2 ln z 1
4 dx
x 22
2 12 1
14
dx
x 22
2 12 1
I 28 ln t
28 ln z
24 arctan 2 x 1 2
4 arctan 2 x 1
I 28 ln x
2 2 x 1 28 ln x
2 2 x 1 24 arctan 2 x 1 2
4 arctan
1296
dxx4x21
x21x4x21x21
dx x21x61
dx x21x31x31
dx
x21x31x31
ax2bxcx31
dx2exf
x31|x3 1x3 1
x2 1 ax2 bx cx3 1 dx2 ex fx3 1x2 1 ax5 ax2 bx4 bx cx3 c dx5 dx2 ex4 ex fx3 fx2 1 x5a d x4b e x3c f x2a d xb e c fa d 0b e 0c f 0a d 1b e 0c f 11 0 0 1 0 0 00 1 0 0 1 0 00 0 1 0 0 1 01 0 0 1 0 0 10 1 0 0 1 0 00 0 1 0 0 1 1
gauss
1 0 0 1 0 0 00 1 0 0 1 0 00 0 1 0 0 1 00 0 0 2 0 0 10 0 0 0 2 0 00 0 0 0 0 2 1
f 12 e 0 d 12
c 12 0 c 12 b 0
a 12 0 a 1
2
I 12 x2 12x31
12 x2 12
x31 1
2 x21x31
12
x21x31
12
x21x1x2x1
dx 12
x21x1x2x1
dx
I 12
x1x2x1
dx 12
x21x31
dx 2x11x2x1
dx 12
x1x1x1x2x1
dx
I 2x1x2x1
dx dxx2x1
2x11x2x1
dx
I dtt dx
x 122 14 1
2x1x2x1
dx dxx2x1
I ln t dx
x 122 34 dzz dx
x 122 14 1
I lnx2 x 1 2 33 arctan 2x1 3
3 ln z dx
x 122 34
I lnx2 x 1 2 33 arctan 2x1 3
3 lnx2 x 1 2 33 arctan 2x1 3
3
I ln x2x1x2x1
2 33 arctan 2x1
3 2 3
3 arctan 2x13 C
?????????1297 dx
1x2 2
x tan tdx 1 tan2tdt
1tan2tdt
1tan2t2 dt
1tan2t dt
1 sin2tcos2t
dtcos2tsin2tcos2t
cos2tdtcos2tsin2t
cos2tdt1 cos2tdt
12 1 cos2t 12 dt
12 cos2tdt
12 t
14 sin2t
12 arctanx
14 sin2arctanx
1298I 3x5
x22x2dx 3x5
x121dx 3 x
x121dx 5 dx
x121 3 x
x121dx
I 3 x11x121
dx 5arctanx 1 3 x1x121
dx 3 dxx121
5arctanx 1
I 3 x1x121
dx 3arctanx 1 5arctanx 1 3 x1x121
dx 2arctanx 1
I1 x 1 tdx dt
3 tt21dt
t2 1 u2tdt du
3 du2u 32 lnu
32 lnt
2 1I1 3
2 ln x 12 1 CI 3
2 ln x 12 1 2arctanx 1 C
1299 dx
x1x2x12
1
x1x2x12 a
x1 bxc
x2x12 dxe
x2x1|x 1x2 x 12
1 ax2 x 12 bx cx 1 dx ex 1x2 x 11 ax4 2ax3 3ax2 2ax a bx2 bx cx c dx4 2dx3 2dx2 dx ex3 2e
x2 2ex e1 x4a d x32a e 2d x23a b 2d 2e x2a b c d 2e a c a d 02a e 2d 03a b 2d 2e 02a b c d 2e 0a c e 11 0 0 1 0 02 0 0 2 1 03 1 0 2 2 02 1 1 1 2 01 0 1 0 1 1
Gaussova
eliminacija
1 0 0 1 0 00 1 0 1 2 00 0 1 0 0 00 0 0 1 1 10 0 0 0 1 0
e 0d 1c 0b d 2e 0b 1 0b 1a 1 dx
x1x2x12 dx
x1 xdx
x2x12 xdx
x2x1
lnx 1 dxx2x1
dx
x2x12
lnx 1 dxx2x1 14
14 dx
x2x12
I lnx 1 I1 I2
I1 dxx2x1 14
14 dx
x 122 34
I1 x 1
2 t
dx dt
I1 dtt2 34
23arctan 2
3t 2
3arctan 2
3x 1
2 23arctan 2x1
3
I2 dx
x2x12 dx
x 122 34
2
I2 x 1
2 t
dx dt
I2 dt
t2 342
I2 u 1
t2 342 dt dv
256t4t233
dt du t v
I2 t
t2 342 256t2
4t233dt t
t2 342 64 4t233
4t233dt
I2 t
t2 342 64 dt
4t232 3 dt
t2 343 t
t2 342 dt
t2 342 3 dt
t2 343
I2 t
t2 342 I2 3 dt
t2 343 . . . . odusta san