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