1031

5 7

x m x n 2 x

dx 1

x 2m 2x mn x 2n x1

dx x 2m x 2 2x mn x 2 x 2n x 2 dx1 2 x 2m 1 1 2 2m 1 1 2 x 2 mn 1 1 2 mn 1 1

1

1

1

5a 2 x 6 dx 5a 2 x 6 dx 5a 2 x77 1032

a2x7 C3 2

x 2m 2 2x mn 2 x 2n 2 dx 2m 1 2 x 2m 1 2

x

2n 1 1 2

2n 1 1 2

C

6x 2 8x 3dx 6 x 2 dx 8 xdx 3 dx 6 x3 8 x2 3x 2x 3 4x 2 3x C1033

2

mn 1 2 x mn 1 2

2n 1 x 2 2n 1 2

C

1042

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 x21 4

a x 4 ax1

2 dx ax 2 a 2 a x x dx ax 2 a 2 4a 2 x 6ax 4 a x 2 3 3 1 3 3

1

1

a 2 x 2 a 2 4a 2 x 6ax 4 a x 2 x 2 a 23 1 1 1 x 2 1 1 2

a 2 5 x2 5 2 1

1

1 x

a 2 4a 2 6 x a 4x a3 3

3

x4

1 3

bx 3

1 3

ax 3

1 2

abx 2 C

1034

a 2 4a 2 x 6a 2 x 2 4 x2 33

3

3

2

a 5

a 2 2a 2 x 4a 2 x 4ax 2 2x

a bx 3 2 dx a 2 2abx 3 b 2 x 6 dx a 2 x 2ab x44 b 2 x76 a2x 1 2

2a 2 x 4ax 4 a x 2 2x 2 1043

2 5 a

x2 C

abx 4

1 7

b2x6 C

1035

dx x 2 7

1 7

arctan

1 7

xC1 10

2px dx2px t| t 1036dt 2p

1044dx x 2 10

1 10

arctanh

xC

2pdx dt 1

10451 2p

t 2 dt x n n1n n n 1 n

1

1 2 2p 3

t2 n1 n n1 n

3

1 3p

t2 n1

3

1 3p

2px 2

3

2 3

2 x px C

dx 4x 2

arcsinh

1 2

xC

1046dx 8x 2

n

dx x

x n dx

x

n n xn1 C

arcsinh

1 4

2xC

1037n nx 1n dx

10472x 2 2x 2 4x 2 1 2

nx u ndx dun u 1n

dx 2x

2x 2 2x 2

arcsinn u 1n du

2x 2 2x 2 arcsinh 1 2 2

dx xC dx

1 2x 2

dx

1 2x 2

dx

du n

2

1 n

1 u n n n 1n nn n

1n

n

1 un n 1n

1

u n nx n C

1

1

1048*a

tan 2 xdx 1048*b

1038

sin 2 x cos 2 x

dx

1cos 2 x cos 2 x

1 cos 2 x

dx dx sinh x cosh x

1 cos x

sin x x C

a 3 x 3 3 dx a 2 3a 3 x 3 3a 3 x 3 x 2 dx a 2 x 3a 10394 3 5 x3 5 3

2

4

2

2

4

3a

2 3

7 x3 7 3

x3 3

a2x 3 2

9 5

a x

4 3

5 3

9 7

a x

2 3

7 3

tanh 2 xdx 1 3

x3 C

sinh 2 cosh 2

1cosh 2 x cosh 2

dx cosh 2

dx

x tanh x x C

1049a

cot 2 xdx 2 55 2

x 1x x 1dx x 1 dx x x C1040

cos 2 x sin 2 x

dx

1sin 2 x sin 2 x

dx

dx sin 2 x

dx cot x x C dx coth x x C

1049b

coth 2 xdx 3 x2

x 2 1x 2 2 3 13

cosh 2 x sinh 2 x

dx

1sinh 2 x sinh 2 x

dx

dx sinh 2 x

dx 3 77 3

x 4 x 2 2 x32

dx x

10 3

x 2x

4 3

2 3

dx

3 13

x

13 3

3 7

x 2 3x

7 3

1 3

1050

x

13 3

x 63 x C

3 x e x dx 3e x dx

1041

3e x u 3 e 1 ln 3dx du 3e x 1051**du 3 x e x 1ln 3 x x

x1 du 1ln 3

1 1ln 3

u

3e x 1ln 3

C

2 xdx dx 2 1057

dx x1

x2 2

x 2 lnx 1 C dx 7 dx x3

adx ax

a

dx ax

a x u| dx du a 1052**du u

x 3dx 9

x 2 5x7 x3

dx

x2 x3

dx 5 dx x3

5 dx 15

x x3

x 2 99 x3

dx 5

x33 x3

dx 7 lnx 3

a ln u a lna x C2x12 2x1

x2 3x 9 lnx 2 1 2 x 2x lnx 2

dx x3

7 lnx 3

3 5x 15 lnx 3 7 lnx 3 3 C dx

10582 2x1

2x3 2x1

dx

dx dx

2 2x1

dx x

dx

2x 1 u| 2dx du x 1054du u

x 4 x 2 1 x1 x 4 1 x1

dx dx

3 lnx 1

x ln u x ln2x 1 C

3 lnx 1 x 1dx x 1x 1dx2

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

x4 x1

1

x 2 11 x1

dx lnx 1

xdx bxdx 1 abx b abx dx x a abx b b

1 b

bxaa abx

dx

1 b

bxaa abx

dx

1 b

dx

a b

dx abx

3 lnx 1 x2 x x 3 x 2 x 1dx 3 lnx 1 1 x 4 1 x 3 x 2 2x 3 lnx 1 C 4 3 1059

x2 2

x

x4 4

x3 3

x2 2

x

a bx u| bdx du x b

a

b xa

dx a 2 dx 2

2

ba xa

dx

b2 xa 2

dx a 2 x 2ab lnx a b 2

dx xa 2

x a t| b a

a b

du b

u

x b

a b2

du u

x b

a b2

ln u

x b

a b2

lna bx C

dx dt a x 2ab lnx a b 2 2

1055

xa 2 2ab lnx a

b2 t

dt t2

xa 2 2ab lnx a dx x1

b2 xa

Cdx x1 2

x axb xdx b a x dx a x x dx x x x dx dx a a dx b x dx b x x x a dx dx a a dx b x ax x dx x a ba dx dx ax ax b x x

1060*

bdx x

x x1 2

dx

x11 x1 2

dx

dx x1 2

lnx 1

x 1 t| dx dt lnx 1 dt t2

x u| dx du ax

lnx 1

1 t

lnx 1

1 x1

C

ba

du

1061 ax

u

ba 2

du u

ax

ba 2

x ln u a

ba 2

lnx C

bdy 1y

b

dy 1y

1056

x 1 : x 1 x 1 x2 x 1x

x 2 1 x1 2

1 y t| 2 x1

dy dt b 1062dt t

2b t 2b 1 y C

a bx dx

a bx t| bdx dt t 1063**dt b

10712 3b

1 t dt b

t

3 2

2 3b

a bx C

3 2

dx 78x 2

1 8

dx7 8

x 2

1 2 2

2 2

arcsinh

2 14 7

x

2 4

arcsinh

2 14 7

xC

1072

x x 2 1

dx

dx 75x 2

5 5

dx7 5

x 2

5 5

arcsin

35 7

xC

x 2 1 t| 2xdx dt 1064x t dt 2x

1073

1 x dt t

1 2

t

x2 1 C

2x5 3x 2 2

dx 2 3

1 3

2x5 x2 2 3

dx

1 3

2x x2 2 3

dx 5

dx x2 2 3

x2 10741 3 1 3

t| 52 3 6 2 5 6

2xdx dt dx

x ln x x

dx

1 x

ln xdx

dt t 2

arctanh

6 2

x1 2

1 3

ln t

5 6

6 arctanh

1 2

x 6

ln x t|dx x

lnx dx 7 5

6 arctanh dx 3 5

x 6 C 1 5

dt1 2 t 2

2 x tdt 2 x 1065

2 x 15 5

1 2

ln 2 x C15 5

arctan15 5

32x 5x 2 7

1 5

32x x2 7 5

dx x2 7 5

2xdx x2 7 5

3 35 35

arctan

35 7

x

1 5

2xdx x2 7 5

dx 3x 2 5

1 3

dx x2 5 3

1 3

arctan

x

15 15

xC

x2 3 35 35 3 35 35

t| 35 7 35 7

2xdx dt arctan arctan dx 1 5

1066dx 7x 2 8

x x

1 7

dx x2 8 7

1 7

1 4

14 arctanh

1 4

x 14

14 28

arctanh

14 4

1 5 1 5

dt t

3 35 35 7 5

arctan

35 7

x

1 5

ln t

xC

lnx 2 dx

C

1067dx ababx 2 ab ab ab

1075 1 ab

dxab ab

x 2 ab ab

1 ab 1 a 2 b 2

1ab ab

arctanh

xab ab

C dx 1 a a3 a 2 x 2

3x1 5x 2 1

5 5

3x1 x2 1 5

3 5 5

x x2 1 5

dx

5 5

dx x2 1 5

1068

arctanh

x ab ab

arctanh

x a 2 b 2 ab

x2 3 5 5 3 5 10

t| 5 5

2xdx dt2 2

x2 x 2 2

dx

x 2 22 x 2 2

dx dx 2 dx

dx x 2 2

x 2 arctan x a

xC dxx a

dt 2

t

arcsinh 5 x 5 5

3 5 10

dt t 2

5 5 1 5

arcsinh 5 x 5 5

1069x3 a 2 x 2

2 t dx

arcsinh 5 x 3dx x 2 4

3 5 5

x

arcsinh x 5 C

dx arctanh arctanh dx

a3 10701 a

axa 2 axx 2 axax x a x a

x 3 axax

dx

x 3 a 3 a 3 dx axax x 3 1 a a arctanh a

2

a 2 axx 2 ax

a 3 x 3 axax

1076

a3

arctanhxxa ax 2

x a

x3 x 2 4

xdx x 2 4

a3 1 a

a lna x xdx dx dx 5 xCxdx x 2 4

a 2 axx 2 ax 2

dx a 3

1 a

arctanh1 2

2

a2 ax

dx

dx C

x 2 4 t| 2xdx dtdt 2

x a lna x a arctanhdx x 2 4

x5 2

x 2 5x6 x 2 4 2

x 2 5x42 x 2 4 1 2

2

t

3 ln x x 2 4

1 2

dt t

3 ln x x 2 4 C

lnx 4 arctan

1 2

2 t 3 ln x x 2 4

x 2 4 3 ln x x 2 4

1077

xdx x 2 5

x6 t2| 2 x 3 t| 3x 2 dx dtdt t

1

x 2 5 t| 2xdx dt 1078dt 2

t

1 2

1 2

ln t

1 2

lnx 2 5 C

1083

dt 3

t 2 1

1 3

dt t 2 1

1 3

ln t t 2 1

1 3

ln x 3 x 6 1

C

xdx 2x 2 3

2x 2 3 t 4xdx dt 1079dt 4

dt t

arcsin x 1x 2

dx

arcsin x u| dx 1x 2

du

u du

2 3

u2

3

2 3

arcsin x 2 C

3

t

1 4

1 4

ln t 1 a

1 4

ln2x 2 3 C dx 1 a

1084

ax2

arctan x 2 4

x 2

axb a 2 x 2 b 2 1 a

dx a 0 ax

dx

axb ax 2 ba2

ax 2 ba

dx

b a2

dx x2 b2a 2

arctan2dx x 2 4 1 2

x 2

u| u2 4

2

2 ax 2 ba

dx t|b a2 b2 a

b a2

du 1 4

2 x2 b2 a

udu

arctan 2

ax 10801 a 1 2a

b2 a

x 2

Carctan 2x 14x 2

1085 a b

2axdx dt

dt 2

t

dx

arctan x1 a

lnax 2

arctan

a b a b

1 2a

dt t

1 a

arctan

a b

x

x arctan 2x 14x 2

dx

x 14x 2

dx t2dx 14x 2

dx

xC

1 4x 2 u| 8xdx du du 8

dt1 2

xdx a 4 x 4

u

t1 3

dt 2

1 83 2

ln u

2 3

t2

3

1 8

ln u

1 3

t2

3

x4 t2| 2 x 2 t| 2xdx dt 1081dt 2

1

1 8

ln1 4x 2

arctan 2x C

1086

1 2

dx 1x 2 lnx 1x 2

a 4 t 2

dt a 4 t 2

1 2

arctan

t a 4 t 2

1 2

arctan

x2 a 4 x 4

C

lnx 1 x 2 u| dx x 2 1

du

x2 1x 6

dx1 2

du u

2 u 2 lnx 1 x 2 C

x6 t2| x 3 t|

1087

ae mx dxmx u| dt 1t 2

3x 2 dx dt 1082dt 3

1t 2

1 3

1 3

arctan t

1 3

arctan x 3 C 1088

mdx dua ae u du m e u du m a m

eu

a m

e mx C

x 2 dx x 6 1

4 23x dx

2 3x t| 3dx dt 4t 1089 1 4t 3dt 3 t

4 t u| 4 ln 4dt du1 1 3 ln 4 du 3 ln 4 u 4t 3 ln 4 1 3 ln 4 4 23x C

1095

1 4 t dt 34 t ln 4 du

ex x2

1

dx t| dt1

1 x dx x2 t

e t e t dt 2 e e dt 2 sinh tdt 2 cosh t C 2 e t e t dt e t dt e t dt e t e t Ct t

e dt e t e x C 1096

1090x x x e a e a 2 dx e 2 a dx 2 dx

5

x dx x

dx x e2 a

x t| dx 2 x

x 2 a t| 2dx a

dt2 ln 5

dta 2

5 t 2dt 2 5 t dt

5t

2 ln 5

5

x

C

e t a dt 2x a e t dt a e t dt 2x 2 2 2 x x 1 ae 2 a 2x 1 ae 2 a C 2 2 1091

e t dt

1097

dx

ex e x 1

dx

e x 1 t| dx x a b a 2x 2a x b x b 2x dx axbx x 2x b dx a

a x b x 2 axbx

ax bxa b

dx 2dx x

1092

dx

ln

a b

2x 3

b a

x b a

bx ax

e x dx dt 1098dt t

ln

C

ln t lne x 1 C

a 2x 1 ax

dx a 2x 2 x dx a 2 x dx a 2 x dx a 2 x dx a u 2 du a t 2dt 3 1 x t| 2 dx dt 2 at 2 ln a

1

1

1

e x a be x dxa be x t| be x dx dt tdt b

3 x u| 2 3 dx du 2

1093

2 3 ln a

au

2 ln a

a 3 a x C

3x 2

1 2

1 t dt b1

2 3b

t2

3

2 3b

a be x 2 C

3

1099x x e a 1 3 e a dx

e x 2 1 xdx e x 2 1 xdxx 2 1 t| 2xdx dtdt e t 2 1 e t dt 1 e t 2 2 1 x 2 1 2e C

e a 1 t| 1 a1 3

x

e a dx dt1

x

t adt a t 3 dt 1100*

3 4

at 3

4

3 4

ae a 1 3 C

x

4

1094

dx 2 x 3

x7 x 2 dxx t| 7tdt 2 2

2 x 3 t| 2 x ln 2dx dt dt 2 x ln 2

2xdx dt 1 2

t

19 4

1 ln 2

dt 2xt

t 3 29 4

1 ln 2

dt t3t

1 ln 2

2 3

dt t 2 3t

1 ln 2

dt t 3 2 9 4 2

7 t dt

1 2 ln 7

7t

1 2 ln 7

7x C

2

1 ln 2

arctanh

2 3 ln 2

arctanh

t1

2 3 ln 2

arctanha x dx 1a 2x

2 3

2 x 3 1 C

1107

1101

cos xax u a x ln adx dudx 2 x

dx x

x z dz cos z2dz 2 cos zdz 2 sin z 2 sin x C 1108ln x sinlg x dx sin ln 2 dx x x ln x ln 2 dx x ln 2

1102

du ln a1u 2

1 ln a

du 1u 2

1 ln a

arctan u

1 ln a

arctan a x C

e bx 1e 2bx

dx

e bx u be bx dx du 1103du b

z| dz

1u 2

1 b dt

du 1u 2

1 b

arctanh u

1 b

arctanh e bx C

sinz ln 2dz ln 2 sin zdz ln x ln 2 cos z ln 2 cos ln 2 ln 2 cos lg x C 1109

et 1e 2t t

sin 2 xdx 2x z| x 2

1cos 2x 2

1 2

dx

1 2

cos 2xdx

e u e t dt du 1104du 1u 2

2dx dz 1 2

arcsin u arcsin e t C

cos z 1 dz 21 2

x 2

1 4

cos zdz 1 2

1 2

x

1 4

sin z

1 2

x1 2

1 4

sin 2x C1 2

1110*

sina bxdxa bx z bdx dz sin z 1105dz b

cos 2 xdx 2x z| 2dx dz

1 cos 2x 2 dx

dx

1 2

cos 2xdx

x

cos 2xdx

1 b

sin zdz

1 b

cos z

1 b

cosa bx C

1 2

x

1 2

cos z dz 2

1 2

x

1 4

cos zdz

1 2

x

1 4

sin z

1 2

x

1 4

sin 2x C

1111x 2 x 2 dx 2

cos

dx z dz 2 cos zdz 2 2

sec 2 ax bdx ax b t adx dt 2 sin z 2 sinx 2

dx cos 2 axb

cos z 2 dz 1106

C

dt a

cos 2 t

1 a

dt cos 2 t

1 a

tan t

1 a

tanax b C dx dx sin 2 ax

1112

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 2axdx2

cot 2 axdx ax t adx dt dt a

cos 2 ax sin 2 ax

dx

1sin 2 ax sin 2 ax

dx

2ax z 2adx dzdz x sin z 2a x 1 2a

sin 2 t

x

1 a

dt sin 2 t

x

1 a

cot t x

1 a

cotax x C

sin zdz x

1 2a

cos z x

1 2a

cos 2ax C

1113

dx x sin a x a dx a

1119 t dt adt sin t

tan xdx a ln tant 2

sin x cos x

dx

1cos 2x 1cos 2x

dx sin 2x 1cos 2x

1cos 2x 1cos 2x

1cos 2x 1cos 2x

dx

adt sin t

a ln tan

x a

2

a lntan

x 2a

C

1cos 2 2x 1cos 2x

dx

sin 2 2x 1cos 2x

dx

dx

1114dx 3 cos5x 4

1 cos 2x t| 1120 dt 2 t

2 sin 2xdx dt t dx 1 ln t 1 ln1 cos 2x C 2 2 dx 1 2 1 2

5x 1115dt 5

4

5dx dtdt 1 1 t 15 cos t 15 ln tan 2 3 cos t 4 1 1 1 1 lntan 2 5x 4 4 15 ln 15

tan

cot xdx 5 2

cos x sin x

1cos 2x 1cos 2x

dx

1cos 2x 1cos 2x

dx

1cos 2x 1cos 2x

1cos 2x 1cos 2x

dx

1cos 2 2x 1cos 2x

x

1 8

C

sin 2x 1cos 2x

dx

1 cos 2x t| 2 sin 2xdx dt

dx sinaxb

ax b t adx dt 1116dt a

dt sin t

dt 2

t

x ab

1 2

ln t

1 2

ln1 cos 2x C

1121

sin t

1 a

1 a

ln tan

t 2

1 a

ln tan

axb 2

C

cot

dx u| dux ab

xdx cos 2 x 2

x ab dx ab

x2 t 2xdx dt 1117dt 2

cot ua bdu a b cot udu a b lnsin u a b ln sin 11221 2

C

cos 2 t

dt cos 2 t

1 2

tan t

1 2

tan x C

2

dx x tan 5 x 5 dx 5

cot u| du

x 5

dx

x sin1 x 2 dx1x t 2xdx dt sin t dt 1 sin tdt 2 2 11181 2 2

cot u5du 5 sin u t| cos t 1 2

cos u sin u

du

cos1 x 2 C 1123

cos udu dt 5dt t

5 ln t 5 lnsin u 5 ln sindx x

x 5

C

1 sin x 2

1

2

dx

1 sin 2 x 2

2 sin x 2

1 dx x 2

dx sin x 2

dx sin 2 x 2

x 2 t 2 dx dt x 2dt 2

tan xdx 2 x

x u| dt 2 sin 2 t

sin t

x 2 2 2

dt sin t

2 2

dt sin 2 t

x 2 ln tan

t 2

2 2

cot t

dusin u cos u

x 2 ln tan

x 2 2

cot x 2 C

2 tan udu 2 tan u 2

du

cos u t| sin udu dt 2 1124dt t

ax t adx dt 1 a

2 ln t 2 lncos u 2 ln cos x C

cos t sin 5 t

dt

sin t z2

x cotx 1dxx 1 u 2xdx du cot u du 2 11251 2 2 1 a

cos tdt dz

dz z5

dx

1 4az 4

1 4a sin 4 t

1 4a sin 4 ax

C

1129

cot udu 2dx sin 2x

1 2

lnsin u

1 2

ln sinx 2 1 C

sin 3x 3cos 3x

3 cos 3x z| dxsin 2x 2

dx sin x cos x

3 sin 3xdx dz 1130sin 3x dz z 3 sin 3x

2x u 2dx du 2 1126du 2

1 3sin 2x 2

dz z

1 3

ln z

1 3

ln3 cos 3x C

sin u

x 2

du sin u

ln tan

u 2

lntan x C

sin x cos x cos 2 xsin 2 x

dx

cos 2x

dx

1 2

sin 2x cos 2x

dx

cos 2x t| x 2

cos sin dxx 2 dx 2

2 sin 2xdx dt 1 2

u du

dt 2

t

1 4

dt t

1 2

t

1 2

cos 2x C

1131

2 cos u sin udu sin u z cos udu dz 2 zdz z 2 sin 2 u sin 2 1127x 2

1 3 cos 2 x sin 2xdx 2 sin x cos x 1 3 cos 2 x dxcos x t| sin xdx dt C 2 t 1 3t 2 dt 1 3t 2 z 6tdt dz 2 6

sin 3 6x cos 6xdx6x t 6dx dt sin 3 t cos t dt 6 sin t z cos tdt dz 1 6 1 6

z dz 2 z 2 2 1 3t 2 2 2 1 3 cos 2 x 2 C 9 9 9sin 3 cos 3x 3 x 3

3

3

3

sin 3 t cos tdt

1132x x tan 3 3 sec 2 3 dx 1 cos 2x 3

dx

sin 3 1 x 3 cos 5 1 x 3

dx

cos z4 1 24

x 3

t| x 3

z 3 dz dx

1 24

sin t 4

1 24

sin 4 6x C 31 4t 4

3

1 3

sint5

dt 3 x 3

1128

3 1133

x sin 2 3 dt

1t 2 t5 3 2 cos 2

dt 3 x 3

cos ax sin 5 ax

1 2t 2

3 4 cos 4

1 t5

dt 3

1 t3

dt

C

tan x cos 2 x

dx dt2 3

1138

tan x tdx cos 2 x

2 sinh 5x 3 cosh 5xdx 2 sinh 5xdx 3 cosh 5xdx5x t t3 2

t dt 1134 a cot 3 x sin 2 x2

2 3

tan x C 1139 dx

3 2

5dx dt2 5

sinh tdt x

3 5

cosh tdt 1 4

2 5

cosh t 1 2

3 5

sinh t 1 4e 2x

2 5

cosh 5x

3 5

sinh 5x C

dx 5 3

2 cos 3 x 2 sin 3 x sin 2 x

dx

cos 3 x sin x8 3

2

sh 2 xdx e e 2 dx 2x

e 2x

dx

sin x t| 5 3

2x t 2dx dt3 53

cos xdx dtdt5 3

2 3

3 5

2 cos 3 x 8 sin 3 x

cos x

2 3

dt sin x8 3

5

3 5

dt sin x sin x5 3

1 sin x

3 5

dt t sin x

3 5

dt tt3 5

3 5

dt t8 5

1 4

e t dt 1 x 2 22 e x e x

1 4

dt 2

et

2

1 8

et

1 2

x

1 8e t

1 8

e 2x

1 2

x

1 8e 2x

C

5 3t3 5

t 5 sin 3 x

3 5

1140

C

b

2 cot 3 x sin 2 x

dx sinh x

dx

dx

e 2x 1 ex

dx

2e x e 2x 1

dx

cot x t|2 3

ex t e x dx dt 3 5 2dt t 2 1

dx sin 2 x dt

2 arctanh t 2 arctanh e x C 2 e x e x

t dt 1135

t

5 3

3 5

cot x C

5 3

1141

dx cos 2 3x

1sin 3x cos 2 3x

sin 3x cos 2 3x

dx cosh x

dx 2

dxe 2x 1 ex

2

e x dx e 2x 1

3x t 3dx dt 1136dt 3

cos 3x z 3 sin 3xdx dz dz 3 z2

ex t e dx dt1 3 x

cos 2 t

1 3

dt cos 2 t

1 3

z 2 dz dx sin ax

cot t

1 3z

1 3

cot 3x

1 3 cos 3x

C

2 1142

dt t 2 1

2 arctan t 2 arctan e x C dx1 2

cos axsin ax 2 sin ax

dx

12 sin ax cos ax sin ax

dx

2 cos axdx

dx sinh x cosh x x 2 dx 2

cosh

x 2

1

1 2

cosh

x 2

1

dx1 4 x cosh 2 2 1

1 2

dx sinh

x 2

2

dx x sinh 2

ax t adx dt 1 a

u du 2 arctanhe u 2 arctanhe 2 Csinh x cosh xx

dt sin t

2 a

cos tdt dx sin 2 3x

1 a

ln tan

t 2

2 a

sin t

1 a

ln tan

ax 2

2 a

sin ax C

1143

du sinh u

1137

csc 2 3x ba cot 3x

dx

tanh xdx

dx

ba cot 3x

cot 3x t3dx sin 2 x dt

cosh x t sinh xdx dt dt t

dt 3 bat

ln t ln cosh t C

1 3

dt bat

1 3a

lnb at

1 3a

lnb a cot 3x C

1144

coth xdx

cosh x sinh x

dx

1 3 ln t t 2 2 3 1 3

3 3 1 2

1 zdz tz t 1 z 2 2

1 3 ln t t 2 2 31 3 2 2

3 1 2

1 z 2 2z

zdz1 2

sinh x t cosh xdx dt 1145dt t

3 ln t t 2

2

dz 1 2

3 ln t t 2 t2 22 2

2

3 2 arctan

ln t ln sinh x C

1 3 ln t t 2 2 3 1 3 ln 3 1 3

3 2 arctan1 2

2 3x 2 2 3x 2 2 2 3x 2 3 x 3 23x 2

3 2 arctan1 2

2 3x 2 2

3 ln

3 2 arctandx x2 2 3

x 5 5 x 2 dx5 x2 t 2xdx dt 5 t 1146dt 2 5 1 5 t dt 12 t 5 26

2 3xCdx x2 2 3

5 12

3 23x 2 23x 2 1 2

dx 1 2

dx 3 3

dx 23x 2

1 2

3 3

5 x 2 5 C

6

1 2

6 arctan1 2

x 6 1 3

arcsinh1 2

x 6

6 arctan

x 6

3 arcsinh

x 6 Cdx x1

x 4 4x1

x 3 1

1150 dx

x 4 4x 1 t 4x 3 4dx dt 1147dt 4

t

dx

1 4

ln t

1 4

lnx 4 4x 1 C

x3 x 8 5

x4 t 4x 3 dx dt 1148dt 4 t 2 5

x 1 : x 1 x 2 x 1 x3 x2 1 x 2 x 2 x 1 x x1 2 1 3

x 3 1 x1 3

dx x 2 dx xdx dx 2 2 x1

x3

1 2

x 2 x 2 lnx 1 C1

1 4

dt t 2 5

5 20

arctan

t 5 5

5 20

arctan

x4 5

5

C

1151

dx ex

e 2 x dx

xe x 2 dxx 2 u 2xdx du 1149eu 2

1x u 2 dx du 2 2 e u du 2e u 2e 2 x C1 21

du 1 e u du 2 dx2 2

eu

1 2

e x C

2

1152

23x 2

1sin x xcos x

dx

3 23x 2

x cos x u 1 sin xdx du du u

2 3x t 3t tdt t 2 3x 2

3xdx tdt 1 3 2

ln u lnx cos x C dx

3t t2t 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

1153

t 2 z

tan 3xcot 3x sin 3x

tdt zdz

3x u 3dx du

1 3

tan ucot u sin u

du

1 3

sin u cos u

u cos u sin sin u

du

1 3

1 cos u

1 3

cos u sin 2 u

xdx 1x 4

sin u t cos udu dt 1 3 1 ln 1sin u 1 dt cos u 3 3 t2 1sin 3x 1 3 sin 3x C cos 3x

x 2 u| 2xdx du

1 3

ln

1sin u cos u

1 3t

1 3

ln

1sin u cos u

1 3 sin u

1160

du 2 1u 2

1 2

arcsin u

1 2

arcsin x 2 C

ln

1154

dx x ln 2 x

tan 2 axdxax u| adx du1 t

ln x tdx x

dt 1 ln x

1155

dt t2

C

1 a

1 a

tan 2 udu 1 2

1 a

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

tan ax

ax C

1161sec 2 x

tan 2 x2

dx

dx cos 2 x

tan 2 x2

x sin 2 2 dx x 2 dx 2

tan x tdx cos 2 x

t dt1 2

dt ln t t 2 2 ln tan x tan 2 x 2 C 1162dx 2x 2 1 2 2x 2 1

1156

dt t 2 2

2 sin 2 tdt 2

1 cos 2t 1 cos 2t t

1 2

sin 2t

x 2

1 2

sin x C

2

x 2x 2 1

dx

x 2 2x 2 1

dx

sec 2 xdx 4tan 2 x

2x 2 1 t 4xdx dt 1157dx x2 1 2

tan x tdx cos 2 x

dt arcsint 2

dt 4t 2

2 arctan x 2

1 4t

2 arctan x 2

1 42x 2 1

C

1163

dt 4t 2

arcsin

tan x 2

C

a sin x cos xdxsin x t cos xdx dt a t dt 11581 ln a

dx x cos a x a dx a

t dt a lnsec t tan t a lnx 1sin a x cos a

at

1 ln a

a sin x C

a 1164

dt cos t

C

x23 x 3 1

dx

3

1ln x x

dx dt3 43

x 3 1 u| 3x 2 dx du 1159du 33u

1 ln x tdx x

1 2

3

u2

1 2

3

x 3 1 2 C

t dt 3

t4

3 4

3

1 ln x 4 C

1165

tan x 1dx 2 x1

dx x1

2t 1 t 2 z 21 tdt dz

1 t2 u 2tdt du

x1 t dt C 2 tan tdt 2 lncos t 2 ln cos x 1 1166

xdx sin x 2

2 dz du 2 ln z 2 ln u 2 2 2z 2u x x x ln2t 1 t 2 ln1 t 2 ln2 tan 2 1 tan 2 2 ln1 tan 2 2 C sin xcos x sin xcos x sin xcos x sin 2 x2 sin x cos xcos 2 x sin xcos x dx sin xcos x sin xcos x dx dx sin 2 xcos 2 x12 sin x cos x sin 2 xcos 2 x

dx

1sin 2x cos 2x

dx

1sin 2x cos 2x

dx

dx cos 2x

tan 2xdx

x2 t 2xdx dt 1 2

2x t 2dx dt1 2 dt 1 cos t 1 tan tdt 1 ln 1sin t cos t 2 2 2 1sin 2x ln cos 2x 1 lncos 2x C 2 1 2 1 2

dt sin t

ln 1cos t sin t dx e arctan x 1x 2

1 2

ln

1cos x 2 sin x 2

lncos t

C dx dx 1x 2

1167

11691x 2

e arctan x x ln1x 2 1

dx dt

x ln1x 2 1x 2

arctan x udx 1x 2

ln1 x 2 t2xdx 1x 2 1 2 t 4

1sin sin x 2

x 2 x 2

2

dx t

du

e

arctan x

e u du 1 tdt arctan x e u 2 1 ln 2 1 x 2 arctan x C 4

arctan x 1170

dx 2

dt1sin t 2 sin t

1168

2

2

1sin t 2 sin t

2 2 ln

sin xcos x sin xcos x

dx 2t 1t 2 1t 2 1t 2 2dt 1t 2

2 ln 1cos t 2t cos t sin tx2 x 2 2

12 sin tsin 2 t sin t 1cos x2

dt 2x 2

2

dt sin t

2 dt sin tx 2

sin

x 2

cos

C

sin x cos x dx 1t 2 1t 2

dx

x 2 22 x 2 2

dx dx 2 dx x 1x 2 dx x1x 2

dx x 2 2

x 2 arctanhdx 1x 2

1 2

x 2 C

1171t 2 2t1 2t1t 2 1t 2

2t 1t 2

2t1t 2 1t 2

2t 1t 2 1t 2 1t 2 t 2 2t1

2dt 1t 2

2

dt

atb 2t1t 2

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 ac 0 b 2c d 1 a c 2d 2 b d 1 a 1, d 0, c 1, b 1

ctd 1t 2 2

1x 2 dx x1x 2 x dx 1x 2

x 2 2x1 x1x 2

dx 2

dx x1x 2

2 arctan x

1 x2 t 2xdx dt dt 2

t

2 arctan x a x

dx x1x 2

1 2

ln t 2 arctan x

dx x1x 2

1 x1x 2

bxc 1x 2

t 2 2t1 2t1t 2 1t 2

dt

t1 2t1t 2

dt

t 1t 2

dt

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 I 1 a dx 1x 2 dx a ln x 1x 2 dx ln x x I 11721 2

1 2

ln1 x 2

ln1 x 2 arctan x ln x

2

1 2

ln1 x 2 arctan x ln x C

2

e sin 2 x sin 2xdx 2 e sin 2 x sin x cos xdxsin x t 2 sin x cos xdx dt e dt et sin 2 x 2

2 1178

dt 2a

sin t

1 a

dt sin t

1 a

ln

1cos t sin t

1 a

ln

1cos 2ax sin 2ax

C

C5 43x 2

sin 2t 0 dt sin 2t 0 dt T T2t T 2 T

0 x dt dx1 T 2

1173

53x 43x 2

dx

dx

3x 43x 2

dx

5

: 3

43x 2 : 3

dx

3x 43x 2

dx

T 2

sin xdx

cos x

T 2

cos 2t 0 C T

4 3x 2 t 6xdx dt 11745 3 5 3

1179

1 2 3dt 6

dx x4ln 2 x

dx4 3

x 2

t

5 3

arcsin5 3

1 2

3x1 2

1 2

dt t

ln x tdx x

3 arcsin

3x t

3 arcsin

3 x 4 3x 2 C

dt dt 4t 2

1180

xdt x4t 2

1 4

ln

2t 2t

1 4

ln

ln x2 2ln x

C

dx e x 1

ex 1 t e dx dt dt ext x

dt t 2 t

arccos

x 2

1x 2

dx arccosx 2

t1 4x 2 1 2

1 2

dt t1t

dt t 1 22

1 4

1 2

t

u 1 2 1 21 4

1

x 2

2

dx 1 2 t 2

dx dtx 2

dt dudu u2 1 4

tdt lnu u1 2 1 2

arccos 2

C

ln

u u 1 2

1 2 1 2

ln2u 1 ln2u 1

11811 e tan x sec 2 xdx e tan x cos 2 x dx

ln2t 1 ln2t 1 ln2t 2 ln 2t ln2e x 1 2 ln 2e x 1 C 1175

tan x u1 cos 2 x dx du

dx ababx 2

1 ab

dx 1x 2

1 ab

arctan x C

e u du e u e tan x C 1182

1176ex e 2x 1

dx

sin x cos x 2sin 4 x

dx

ex t e x dx dt 1177dt t 2 1

sin 2 x t 2 sin x cos xdx dt ln e x e 2x 1 C 1183 dxsin 2ax 2

dt 2 2t 2

ln t t 2 1 2

1 2

dxsin 2 2x 4

dt 2t 2

1 2

arcsin

2 2

t

1 2

arcsin

2 2

sin 2 x C

dx sin ax cos ax

dx sin 2ax

dx sin 2 x cos 2 x

4

dx sin 2 2x

2ax t 2adx dt

2x u 2dx du

4 1184

du 2

sin 2 u

2

du sin 2 u

2 cot u 2 cot 2x C x 1x 2 2

ln x x 2 11 x x 2 1 1 x x 2 1 1 x 2 1

t dx dt dx dt

arcsin xx 1x 2

dx

arcsin x 1x 2

1

2x 2 x 2 1 x 2 1 x x 2 1

arcsin x udx 1x 2

du1 2

1x t 2xdx dt 1 2

dx dt2 3

udu 1185

dt t

u2 t

1 2

arcsin 2 x 1 x 2 C

t dt 1189

2 3

t

3 2

ln 2 x x 2 1

3

C

sec x tan x sec 2 x1

dx

sin x 1 cos x cos x 1 cos 2 x

1

dx

sin x cos 2 x 1cos 2 x cos 2 x

dx

sin x cos x

1cos 2 x

dx

sin x cos x 1cos 2 x

dx

x 2 chx 3 3dxx3 3 t 3x 2 dx dt 1 3

1 cos 2 x t| 2 1 cos 2 x t 2 2 sin x cos xdx 2tdt 1186sin x tdt cos xt sin x cos x

cosh tdt dx dt1 ln 3

1 3

sinh t

1 3

sinhx 3 3 C

1190 dt t 2 1

dt cos 2 x

arctanh t arctanh 1 cos 2 x C

3 tanh x cosh 2 x

tanh x t dxdx cosh 2 x

cos 2x 4cos 2 2x

2x t 2dx dt 1 2

3 t dt 1 2

3t

1 ln 3

3 tanh x C

cos t 4cos 2 t

cos t 51cos 2 t

1 2

1192cos t 5sin 2 t

x2x 5 10 dx2x 5 t 2dx dt

sin t u cos tdt du 5 10 1 2

du 5u 2 5 5

5 10

arctanh

5 5

u

5 10

arctanh

5 5

sin t

1 2

t 10 dt dx

1 11 t 22

1 22

2x 5 11 C

arctanh

sin 2x Cdx cos 2 x

1191a

1187*

:cos 2 x dx sin 2 xcos 2 xcos 2 x :cos 2 x

x x 2 2

dx 1cos 2 x

tan 2 x2

1 cos 2 xtan 2 x2

dx

x2 2 txdx x 2 2x 2 2 x

tan x tdx cos 2 x

dt 2 2 dt x2

dt 2 2

arctant 2 2

dt

1188

dt t 2 2

2 2

arctan

tan x 2 2

C

ln x x 2 1 x 2 1

x x 2 2 2 arctan 2 dx 1 x dx x2

dt t 2 2

2 2

arctan

2 2

t

x2 2 C

x x 2 2

dx

t dt

x 2 dt x 1 2 t2

1 t

dt

2t 2 t2

dt 2t 2

arcsin

2 2

t arcsin

2 2x

C

3

1x 1 x

2 x dt 2 2

1t 2 1t

tdt 2 2 t1

tt 3 1t

dt

t t : t 1 t t 2 t3 t2 t 2 t t 2 t 2t 2t 2 2 2 t 2 dt tdt 2 dt 2 2 3 3 2

1191b

dx e x 1

x ln t dx dt t dt t e ln t 11

7

dt t 1 t

1

dt t 1 1 t

dt 1t

ln1 t ln1

1 ex

C

1191c

dt t1

2 3 t 3

t 2 4t 4 ln1 t

x5x 2 3 dx5x 2 3 t 10xdx dt x5x 2 3 7 1191ddt 10x

x x 4 x 4 ln1 x C

1194

1 10

dx x 2x1

t 7 dt

1 8 t 80

1 80

8 5x 2 3 C

2x 1 t 2x 1 t 2dx 2x1

dt dt dt x

x

t 2 1 2 dtt 2 1 2

xdx x1

dt2 2 3 t 3

2x1 x 2x1

2

dt t 2 1

2 arctanh t 2 arctanh 2x 1 C

x1 tdx 2 x1

1195 2t 2 3

2 xdt 2 t 1dt 1191e

x1 2 x1 C

3

dx e x 1

ex 1 te x dx 2 e x 1

cos xdx 1sin 2 x

dt 2dt ex

t sin x dt cos xdx 1192dt 1t 2

1196

2 e x 1 dt ex e x 1

2

dt t 2 1

2 arctan t 2 arctan e x 1 C

arcsinh t arcsinh sin x C

ln 2xdx x ln 4x

t

ln 2xdx xln 2xln 2

ln 2x tdx x t5 2

x2x 5 10 dx2x 5 t x 2dx dt 1193t5 10 dt t 2 2

1197

txdt xtln 2

dt ln 2 1 12 t 48

tdt tln 2 dt tln 2

tln 2ln 2 tln 2

dt

t ln 2 lnt ln 2 ln 2x ln 2 lnln 2x ln 2 C

1 4

t 5t 10 dt

5 11 t 44

1 48

2x 5 12

5 44

2x 5 11 C

arcsin x 2 1x 2

dx

1x 1 x

dx x t

arcsin x tdx 1x 2

dt1 3 t 3

dx 2 x

dt

t 2 dt

1 3

arcsin 3 x C

1198

cos t ue 2x e x 1

sin tdt du 1203du u

ex 1 te x dx 2 e x 1

1 2

u du 2 u 1 u 2 2 cos t 1 cos t 2 3 31 3

3

3

dt dt 2 t 2 1t t

2 2 x2 dt 2 t 2 1dt 2 3 t 3

2 x 2 2 C

3

2 3

e 1

x e 2x 2 e 1 t ex 3 x

2t

2 ex 1 C

x 2 a 2 x 2

dx

1199

x a 2 sin 2 t dx cos x t 2 sin x x dx dt cos xdx sin t cos tdt sin t sin t cos tdt x x

sin 3 x cos x

sin 2 t cos tdt x2

sin 2 t cos tdt sin 2 ta 2

sin t z2 5 t 5

2 sin 2 xdt 2 1 cos 2 xdt 2 1 t 4 dt 2 5

2t

cos tdt dz 1204*z 2 dz z 2 a 2

cos x 2 cos x C

5

dz a 2

1200*

x 2 a 2 a arctandx x x 2 1

dz z 2 a 2 x 2 a 2 a

z a arctan C

z a

sin t a arctan

sin t a

dx x 1x 2

2

1 t

1x

t|

2

1 x2 t2 xdx tdt 1201tdt x

x1 t dt t2 1 t2

dx dt t21 t 2 1

xt

dx

1 x2

dt

dt arctanh t arctanh 1 x C

2

1

dt t2 1 t 1t 2 t2

dt t 1t 2 t2

dt t 1 t

1t 2

dt 1t 2

arcsin t

arcsinx2

1 x

C dx

1x 2

1205

x cos t dx sin tdt 12021 2 cos 2 t 1cos 2 t

cos 2 t sin t sin t

x 2 1 x

x sinh t dt cos 2 tdt 1cos 2t 2

sin tdt 1 4

1 2

t

1 4

sin 2t

dx cosh t sinh t cosh t sinh t cosh t cosh tt sinh t1 sinh t sinh sinh t cosh t 2 arctanhe t cosh arcsinh x 2 arctanhe arcsinh x C2 2

arccos x dx2

sin2 arccos x C

sinh 2 t1

sinh 2 t1

dt sinh t

x3 2x 2

1206*

2 x cos t 2xdx sin tdt x3 cos t sin t cos t sin tdt 2x

x2 cos t sin tdt 2

dx x 2 4x 2

x 2 sin t 2cos t sin tdt 2 cos t

1 2

2 sin tcos t sin t 2 cos t

dt

dx 2 cos tdt

dt

cos t sin tdt

2 cos tdt 2 sin t 2 42 sin t 2 2 cos t 8 sin 2 t cos t

1 4

2 cos t 4 sin 2 t 44 sin 2 t dt sin 2 t

dt

2 cos t 4 sin 2 t 44 sin 2 t 1 4

dt x 2

2 cos t 8 sin 2 t 1sin 2 t

dt

2 cos t 8 sin 2 t cos 2 t

dt

1 4

tan t

tanarcsin

C

dx x 2 4x 2

12101 t dt t2

x1 t2 dt t2

x2 x 2 a 2

dx

dx

x a cosh tdt 41 t2

1 4

4

1 t2

dt1 t

4t 2 1

1 2

t t2 1 4

dt

1 2

1 2

4t 1

2

dx a sinh tdt a 1211a 2 cosh 2 t a 2 cosh 2 ta 2 1sinh 2 t dt sinh t

dt ax a

a 2 cosh 2 t

4t 2 11 4

1207

4 1 1 x1x 2 1x 2

2

1 4x

4 x2 C

a cosh 2 t1 dt a sinh t

dt a arccosh 2x a

cosh 2 t cosh 2 t1

dtt 2 1 2

sinh tdt a cosh t a ln tanh x a lntanh

a cosharccosh

a ln tanh

arccosh

x a

C

1 x 2 dx 1x 2 1x 2 1x 2 1x 2 1x 2 1x 2

dx2 c 1x 2 c 1x 2 c 1x 2

ln xdx|

ax b 1 x ax b 1 x 2 a 1 x2 2xaxb 2 1x 2

dx dv ln x u xvdx x dx x

du x ln x x C

| 1 x2

x ln x x 1212

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 1 , b 0 2 a c 1. . . . c 1 2 I 12081 2

arctan xdxarctan x u dx dvdx 1x 2 1 2

du

xvx 1x 2

x 1 x2

1 2

dx 1x 2

1 2

x 1 x2

arcsin x C

x arctan x 1 x2 t 2xdx dt x arctan x

dx

dx x1x

dt 2

x sin t dx 2 sin t cos tdt 2 sin t cos t sin 2 t1sin 2 t

2

t

x arctan x

1 2

ln t x arctan x

1 2

ln1 x 2 C

1213sin t cos t sin t 1sin 2 t

dt 2

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

arcsin xdxarcsin x u dx dvdx 1x 2

2 arcsin x C 1209

dux

xv1x 2

a 2 x 2 dxx a sinh t dx a cosh t 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 x x 1 a 2 sinh 2t 1 a 2 t 1 a 2 sinh2 arcsinh a 1 a 2 arcsinh a 4 2 4 22

x arcsin x 1 x2 t 2xdx dt x arcsin x a2 2

dx

dt 2

t

x arcsin x

1 2

dt t

x arcsin x t x arcsin x 1 x 2 C

dt

1214

x sin xdx

1 2

x x 2 a 2

1 2

a 2 arcsinh

x a

C

xu dx du 1215

sin xdx dv cos x v

tu dt du

e t dt dv et v

x cos x cos xdx x cos x cos xdx x cos x sin x C

x cos 3xdx3x t 3dx dt 1 9

te t e t dt te t e t 1 1 1 2 2 I 27 t 2 e t 2I 1 27 t 2 e t 2te t e t 27 t 2 e t 27 te t 27 e t 2 3x 1 2 3x 2 e 3x 1 x 2 e 3x 2 xe 3x 2 e 3x C 27 3x e 27 3xe 27 3 9 27 1219

x 2 2x 5e x dx x 2 e x dx 2 xe x dx 5 e x dxx2 u 2xdx du xu1 9

t cos tdttu cos tdt dv sin t v cos3x

e x dx dv e x v

dt du

I 1 e x x 2 e x 2xdx e x x 2 2 xe x dx e x dx dv e x v dx dux

1 t sin t sin tdt 9 1 t sin t 1 cos t 1 3x sin3x 9 9 9 1 x sin 3x 1 cos 3x C 3 9 1216

x ex

dx xu dx du e x dx dv e x vx

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 1220*

xe 1217

x

e dx xe x e x C

x3e 3x 3 t

x

x2 x dxxu 2 x dx dv1 ln 2 1 dx du ln 2 2 x v 1 1 1 ln 2 x2 x ln 2 2 x dx ln 2 x2 x

dx dt 3 27t 3 e t 3dt 81 t 3 e t dt 81I 1

2 x dx

1 ln 2

x2 x

1 ln 2 2

2 x C

I 1 t 3 e t dt e t dt dv et v3 t t 2

1218

t3 u 3t 2 dt du

x 2 e 3x dx3x t 3dx dt t2 9

t e 3 e t dt t 3 e t 3I 2 I 2 e t t 2 dt1 27

et

dt 3

t 2 e t dtt2 u

e t dt dv et v2 t

t2 u 2tdt du

e t dt dv et v I I 1 e t tdt1 27

2tdt du

I 2 t e 2 e t tdt t 2 e t 2I 3 I 3 e t tdt e t dt dv et v tu dt du

t 2 e t 2 e t tdt

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 x x x 81 3 e 3 243 3 e 3 486 3 e 3 486e 3 3x 3 e 1 3

ln x u x 2 dx dvdx x

du

x3 3 x 3 dx 3 x

v x3 3

x3 3

ln x

ln x

1 3

x 2 dx

1 3

x 3 ln x

1 9

x3 C

x

27x 2 e 1 2

1 3

x

162xe

1 3

x

486e

1 3

x

C

1224

1221

ln 2 xdx x sin 2xdxln 2 x u2 ln x x 2

x sin x cos xdx 2x t 2dx dt 1 2

dx dv xv dx x ln 2 x 2 ln xdx

dx du2 ln x x

x ln x xdt 2

sin tt 2

1 8

t sin tdt

I 1 ln xdx ln x u dx dvdx x 1 8 1 8 1 8

ut1 8

dv sin tdt sin t

du dt v cos t t cos t cos tdt t cos t cos tdt t cos t 1 2x cos2x 1 sin2x 1 x cos 2x 1 sin 2x C 8 8 4 8 1222*

du

xv

I 1 x ln x x dx x ln x x x I x ln 2 x 2x ln x x x ln 2 x 2x ln x 2x C 1225

x 2 5x 6 cos 2xdx2x t 2dx dt t2 4

ln x x3

dxdx x

ln x u x 3 dx dv dux 2 2 x 2 2 x 2 dx 2 x

v 1 2ln x x2

3 sin t

5 t 2 1 8

6 cos t

t 2 cos tdt

dt 2

5 4

t cos tdttu dt du

1 8

t cos tdt 2

5 4

t cos tdt 3 cos tdt

ln x 1226

1 2

dx x3

1 ln x 2 x2

1 4x 2

C

t2 u 2tdt du

cos tdt dv sin t v

cos tdt dv sin t v

ln x x

dxdx x

I a t 2 sin t 2 t sin tdt I b t sin t sin tdt t sin t cos t I aa t sin tdt tu 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 1 I a 5 I b 3 sin t 1 t 2 sin t 2t cos t 2 sin t 5 t sin t cos t 8 8 4 4 sin t 1 t 2 sin t 1 t cos t 5 t sin t 5 cos t 8 4 4 4 11 sin2x 1 2x 2 sin2x 1 2x cos2x 5 2x sin2x 5 cos2x 4 8 4 4 4 11 sin 2x 1 x 2 sin 2x 1 x cos 2x 5 x sin 2x 5 cos 2x C 4 2 2 2 4

ln x udx x

dvdx x

du 2 x v 2 x ln x 2 dx x

2 x ln x 2 x 1227

2 x ln x 4 x C

x arctan xdxarctan x u xdx dvdx 1x 2

du1 2

x2 2

v 1 2 x2 2

11 4

x2 2 x2 2

arctan x arctan x

x 2 dx 2 1x 2

arctan x 1 2

1 2

dx

x2 1x 2

dx 1 2

x2 2

arctan x 1 2

1 2

x 2 11 1x 2

dx

dx 1x 2

x 2 arctan x

x

arctan x C

1228

1223

x arcsin xdx

x 2 ln xdx

xdx dv arcsin x u1 2

e x sin xdxe x dx dv ex v dx 2 1 2 1 4

x2 v

dx 1x 2 1 2

du x2dx 1x 2 1x 2 1 1x 2

sin x u cos xdx du

1 2 1 2 1 2

x 2 arcsin x x 2 arcsin x x arcsin x 2 1 2 1 4

x 2 arcsin x arcsin x C

1 2

1 x 2 dx

1 2

dx 1x 2

I e x sin x e x cos xdx I 1 e x cos xdx e x dx dv ex v cos x u sin xdx du

x 1 x

1229

ln x 1 x 2 dxdx dv ln x 1 x 2 xv x ln x 1 x 2 x ln x 1 x 1 x2 t 2xdx dt x ln x 1 x2 2 dx 1x 2

e x cos x e x sin xdx u I 1 e x cos x e x sin xdx I e x sin x e x cos x e x sin xdx 2I e x sin x e x cos x x x I e sin xe cos x C 2 x e sin xdx ex u e x dx dux

dudx 1x 2 x 1x 2

e x sin x e x cos x I

x

x ln x 1 x 2 dx

x 1x 2

dx

sin xdx dv cos x v

dt 2

e cos x cos xe x dx e x cos x cos xe x dx 1 x2 C ex u e x dx dux

t

x ln x 1 x 2 1230

t x ln x 1 x 2

cos xdx dv sin x v

xdx sin 2 x

xu

dx sin 2 x

dv

dx du cot x v x cot x cot xdx x cot x cot xdx x cot x lnsin x C 1231 Ix cos x sin 2 x cos x sin 2 x

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 I e cos xe sin x C 2 1233

3 x cos xdx3 x dx dv3x ln 3

cos x u sin x du

v

dx dv

xu dx du

1 sin x v

3x 3x ln 3 cos x sin x ln 3 dx 3x 1 I ln 3 cos x ln 3 3 x sin xdx

I1

cos x sin 2 x

3 dx dv3x ln 3

x

sin x u cos x du3x 1 sin x ln 3 ln 3 3x sin x ln12 3 ln 2 3 x 3 3x cos x ln 2 3 ln 3

sin x t cos xdx dt 1232dt t2

v cos x cos x

I1 sin x

1 sin x

1 t

1 sin x

Ix sin x

3x ln 3 3x ln 3

I

3x ln 3

cos x

3x ln 3

cos x

3x ln 2 3

sin x

1 ln 2 3

3 x cos x

I x

dx

dx sin x

x sin x

ln tan

x 2

C

I 1

1 ln 2 3

sin x

I 1234

3x ln 3

cos x

3x ln 2 3

sin x

ln 2 31 ln 2 3

3x ln 3

cos x

3x ln 2 3

sin x

ln 2 31

ln 2 3 3 x

cos x ln 3sin x ln 2 31

C

1236

x 3 e x 2 dxx 2 t 2xdx dt x3etdt 2x

e ax sin bxdxbx t bdx dt e b sin t dt b ea bat b t at

1 2

x 2 e t dt

1 2

te t dt

1 b

e b sin tdtsin tdt dv

at

tu dt du1 2 1 2

e t dt dv et v1 2

u

e a b dt du cos t v e cos t eat b at b

1 b

a b

e a cos tdtsin t va b2 a b2 a b2

t b

te t e t dt 1 te t 1 e t 2 2 2 2 2 x 2 e x 1 e x 1 x 2 e x 2 2

e x C

2

1237

u dt duat b at b at b

cos tdt dv e sin t aat b at b

e x dxx t| 21 b

a b

e

t ab

I I I I 1 I I 1235

1 b 1 b 1 b 2 a2 b e1 b

e cos t e cos t e cos t cos t a2 b a 2 b 2 b2 eat b

e

t ab

sin tdt

x t2 dx 2tdt 2 e t tdt tu e t dt dv et v 2te t 2e t 2 x ex

e sin t aI e sin t a b2t at b

at b

1 e cos t bsin t

at b

e sin t eabx b

at b

a2 b2

I

ea b

b cos ta sin t a 2 b 2

b cosbxa sinbx a 2 b 2

dt du 1238

b cos bxa sin bx a 2 b 2

e ax C

2 te t e t dt

2e

x

C

sinln xdxln x tdx x

x 2 2x 3 ln xdxx 2 2x 3dx dv ln x ux3 3

dt sin t u cos tdt du

x 2 3x v2 x3 9

dx x x3 3

du x 2 3x dx xx2 2 x3 3

x sin tdt e t sin tdt e t dt dv et vt

1239

x3 3 x3 3

x 3x ln x x 2 3x ln x dx u dv xdx vx2 2 x2 2 2 x 2 1

x 2 3x ln x

x2 3

x 3 dx

3x C

I e sin t e t cos tdt I 1 e t cos tdt e t dt dv et vt

x ln

1x 1x

ln2 x 2 1 x2 2 x2 2

1x 1x

cos t u sin tdt dut t t t

dx du1x 1x 1x 1x

ln ln dx

dx x 2 1

x2 2

ln

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 ln x ln x I e sin te cos t e sin ln xe cos ln x 1 x sinln x 1 x cosln x C 2 2 2 2

dx

1x x2 x 2 1 1x 1 2 x ln 1x x 2 1x

x2 2

ln

1x 1x

x 2 11 x 2 1

arctanh x C

1240

ln 2 x x2

ln 2 x u2 ln x x

dv

dx du v 2 2

dx x2 1 x

arctan 2 x u2 arctan x 1x 2

xdx dvx2 2

dx du2

v x2 2

1 x 1 x

ln x

ln x du

1 2 ln x x x 2 ln2x dx x dv dx x2 v 1 x

dx

x2 2

arctan x

x 2 2 arctan x 2 1x 2

arctan 2 x

x 2 arctan x 1x 2

arctan x udx 1x 2

ln x udx x

x2 1x 2

dx dv

du

x 2 11 1x 2

dx x arctan x vx 1x 2

1 ln 2 x 2 1 ln x 1 dx x x x x 1 ln 2 x 2 1 ln x x x 1241dx x2

1 x

ln 2 x

2 x

ln x

2 x

C

lnln x x

arctan x x arctan x arctan x I 1 I 2 x I 1 1x 2 dx 1 x2 t 2xdx dt

x2 2 x2 2 x2 2

dx arctan 2 x x arctan x arctan x x arctan x 1x 2

arctan 2 x x arctan x arctan 2 x 2 2

dx

arctan x 1x 2

dx

dxdx x

lnln x udx x ln x

dv dx x

du

ln x v ln x lnln x ln x C

ln x lnln x ln x xdxx ln x lnln x ln 1242

dt 1 ln t t 2 I 2 arctan x dx 1x 21 2

1 2

ln1 x 2

arctan x tdx 1x 2

x arctan 3xdx2

dt 1 arctan 2 x 2 x2 C

3x t 3dx dt t2 9

tdt 1 27

arctan t dt 3 du arctan t t3 3

t 2 arctan tdt

1 2 t 1 arctan 2 x 2 2 x2 2 arctan x x arctan x arctan 2 x 1 ln1 x 2 2 2 1 2 x arctan 2 x x arctan x 1 arctan 2 x 1 ln1 2 2 2

arctan t u t 2 dt dvdt 1t 2

1244 I arcsin x 2 dx dv dx dt 1 81

vt 3 dt 3 1t 2 1 t3 3 1t 2

1 27 1 27

t3 3 t3 3

arctan t

arcsin 2 x u 2arcsin x 1x 2

t 3 arctan t

t3 1t 2

dt

vx

dx du1x 2

1 t2 z 2tdt dz 3 1 t 3 arctan t tz dz 81 2t 2 1 1 1 t 3 arctan t 2 tz dz 81 t 3 arctan t 1 z1 dz z 81 2 1 1 t 3 arctan t 1 dz 1 dz 81 t 3 arctan t 1 z 1 z 81 2 2 2 2 1 t 3 arctan t 1 1 t 2 1 ln1 t 2 81 2 2 1 3x 3 arctan3x 1 1 3x 2 1 ln 1 3x 2 81 2 2 1 27x 3 arctan 3x 1 9 x 2 1 ln1 9x 2 81 2 2 2 1 3 1 1 1 x arctan 3x 162 18 x 2 162 ln1 9x 2 C 3

I x arcsin 2 x x2

arcsin x

dx

Iv x

arcsin x 1x 2 x 1x 2

arcsin x u ln z1 1x 2

dx dv

dx du 1 x 2 v1 1x 2

I v uv vdu 1 x 2 arcsin x 1 x 2 Iv x 1x 2

dx 1 x 2 arcsin x x

1243

1 x2 t 2xdx dt

x arctan 2 xdx

1245

1 2

dt t

1 t 2 dt 1 2t 2 t 1 x 2 2 2

1

1

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 2xdx2x t 2dx dt

arcsin x x2

dxdx x2 1 x

arcsin x udx 1x 2

dv vdx 1x 2

1 x2 t 1 x2 t2 xdx tdt 1 arcsin x xdt x2 dx x 1x 2

1 4

t tan 2 tdttan 2 t dv ut

du

tan t t v du dt I I 12481 4 1 4

1 arcsin x 1 x x I1 1246 xttdt x

dt x

x

1 x

dt 1t 2

arctanh t arctanh 1 x 2

ttan t t tan t tdt 1 t tan t t 2 tan tdt tdt 4 t tan t 1 t 2 lncos t 1 2x tan 2x 2x 2 lncos 2x C 2 4 dx dv sin 2 x u

I 1 arcsin x I 1 x

arcsin x arctanh 1 x 2 C

sin 2 x ex dx ex

arcsin x 1x

dx

x t x t2 dx 2tdt I 2t arcsin t 1t 2

e x v 2 sin x cos xdx sin 2xdx du I e x sin 2 x e x sin 2xdx e x sin 2 x I 1 I 1 e x sin 2xdx sin 2x u tudx ex

dt

dvx

arcsin t 1t 2 1 2

2 cos 2xdx du e

v

dt dv

arcsin 2 t v dt du1 2

e x sin 2x 2 e x cos 2xdx I 1 e x sin 2x 2I 2 t arcsin 2 t arcsin 2 tdt cos 2x udx ex

I 1 arcsin tdt2 2 arcsin t 1t 2

I 2 t 1 arcsin 2 t 2 arcsin 2 t u dt du t arcsin 2 t 2 arcsin t udt 1t 2

arcsin 2 tdt

dv

2 sin 2xdx du e x v 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 e x sin 2x 2e x cos 2x 4I 1 e sin 2x2e cos 2x 5 x sin 2x2e x cos 2x I e x sin 2 x I 1 e x sin 2 x e C 5 1249

dt dv tv1t 2 tdt 1t 2

t arcsin t

dt dv

du v 1 t 22 2 dt 1t 2

cos 2 ln xdxcos 2 ln x u arcsin t 1 t dt2

dx dv xv

I 2 1 t arcsin t 1 t2

2 cos ln x sin ln x dx du x

I 2 arcsin t 1 t t I 1 t arcsin 2 t 2 arcsin t 1 t 2 t2

I x cos 2 ln x x2 cos ln x sin ln x dx x cos 2 ln x x2 cos ln x sin ln x dx x x t arcsin 2 t 2 arcsin t 1 t 2 2t2

I x cos 2 ln x sin2 ln xdx x cos 2 ln x I 1 sin2 ln x u2 x

I t arcsin 2 t I 1 t arcsin 2 t t arcsin 2 t 2 arcsin t 1 t 2 2t I 2 arcsin t 1 t 2t 2 arcsin x 1 x 2 x I 2 1 x arcsin x 2 x C

dx dv xv

cos2 ln xdx du

I 1 sin2 ln xdx x sin2 ln x x 2 cos2 ln xdx x

I 1 x sin2 ln x 2 cos2 ln xdx x sin2 ln x 2I 2 I 2 cos2 ln xdx cos2 ln x u 2 sin2 ln xdx du x I2 I2 I1 I12 x

a 2 a 2 sin 2 t a cos tdt a 2 1 sin 2 t cos tdt a 2 cos 2 tdt a2 1 cos 2tdt2

dv dx vx

1 2

t a 2 arcsin

a2 2

a2 2 x a

cos 2tdt a t a sin 2t 1 2 2 1 4 2 1 4

a 2 sin2 arcsin

x a

C

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 5x sin2 ln x2x cos2 ln x 5

1253*

A x2x dx A 4

A sinh t A cosh tdtA 2

I x cos 2 ln x 1250**

C

A A sinh 2 t A cosh tdt A 1 sinh 2 t cosh tdt A cosh 2 tdt sinh 2t A 2

cosh 2t 1

x2 2 x 2 1

dxxdx 2 x 2 1

tAx 2 Ax 2

A 4

sinh 2 arcsin

x A

A 2

arcsin

x A

C

xu

dv

A x2 1 2 Ax 2 Ax 2 Ax 2 Ax 2

1 dx du 21x 2 v 1 1 21x 2 x 21x 2 dx x 21x 2 1 2

ax b A x 2 c a A x2 2xaxb 2 Ax 2

dx Ax 2 c Ax 2

|

| A x2

1 1x 2

x dx 22x 2

arctan x C1 a2

1251*

I

dx 1 2 a2 x 2 a 2 x a13 arctan a

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

x2 2 x 2 a 2

dx

A x 2 aA x 2 xax b c A x 2 aA 2ax 2 xb c aA c A 2a 1 c A aA A1 a b 0, a 1 2 A 2

x at dx adt I I1 a3 1 a3

,c

A 2

arctan arctan

x a x a

1 a2 1 a3

a2t2 2 a 2 t 2 a 2 t2 2 dt t 2 1

adt

1 a3

arctan

x a

a

t2 2 a 4 t 2 1

I 1 x A x 2 2 dt I 1254*1 2

A 2

dx Ax 2

x A x2 dx

1 2

A ln x A x 2

Cdx 9x 2

t 2 t 2 1

dt dv

tu

1 21t 2

x2 9x 2 9x 2 9x 2 9x 2 9x 2

9x 2 9 9x 2

dx 9 x 2 9 c 9x 2 c 9x 2

9 x 2 9 arcsin

x 3

I I I1 a3 1 a3 1 a3

1 21t 2

v dt dux a x a x a

ax b 9 x 2 a 9 x2 1 2 axbx 9x 2

dx| | 9 x2

arctan arctan arctan

1 a3 1 2a 3 1 2a 3

1 21t 2 t

dt

1 a3

arctan

x a

1 2a 3

t 1t 2

dt 1t 2

t 1t 2 arctan t

x a x 1 a 2

x arctan a C

1252**

9 x2 9 x2 9 x2 c 9a a I1 x 2

a9 x 2 ax bx c 9a 2ax 2 xb c x 2 2a xb c 9a 9. . . c 9 9 2 , b0 , c9 2 9 2

a 2 x 2 dxx a sin t dx a cos tdt

9 x2 x 3

dx 9x 2

x 3

x 2

9 x2 x 2

9 2

arcsin9 2

1 3

xx 3

I 9 arcsin

I 1 9 arcsin

9 x2

arcsin

I 1255

9 2

arcsin

x 3

x 2

9 x2 C dx x1 2 4

x 1261 1 2

5 2

lnx 2 3x 4

9 7

arctan

2x3 7

C2x6 10 6 3 x3 2 1

dx x 2 2x5

dx x 2 2x14

arctan

x1 2

C

1256dx x 2 2x

x2 x 2 6x10

dx

x 3 dx x 2 2x11

dx x1 2 1

arctanhx 1 C

x 2 6x106x10 dx x 2 6x10 2x6 dx dx 8 x 2 6x10 x3 2 1

dx

6x10 x3 2 910

dx x 3

dx

x 2 6x 10 t x 3 z 2x 6dx dt dx dz x 3 ln t 8 arctan zdx

1257dx 3x 2 x1

1 3

dx x 2 1 x 1 3 3

1 3

dx x 1 62 1 36 1 3

1 3

x 1 6

2

11 36

2 11

arctan

6x1 11

C

x 3 1262

dt t

8

1258xdx x 2 7x13

x 3 lnx 2 6x 10 8 arctanx 3 C

dz z 2 1

t7 2

xdx x 7 22

49 13 4

x 7 2 x 7 22

3 4

dx

7 2

dx x 7 22

3 4

dx 33x2x 2

tdt33 16

x ln 2 12591 2 1 2 t t2 3 4

7 2

1 2

dx3 2

3 xx 2 2

1 2

dx33 16

x 3 4

2

dx dt dt

x

3 4

7 2 2

dx x 7 22

3 4

7 3

1 2

ln4t 2 3 C

7 3

3 arctan

1 3

2x 7 3

dx dt 12631 2

ln 4x

3

lnx 2 7x 13

arctan 2x7 3 7 arctan 2x7 3 3

t 2

1 2

arcsin

4t 33

1 2

arcsin

4x3 33

C

xdx x2 2 1 dx 6 x2 2 1

dx xx 2

1 2

dx1 4

x 1 2

2

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

2

dx

dx x2 2 1 dx 2 x2 2 1

x 1264dt1 4

t arcsin 2t arcsin2x 1 C

dx dtt 2

x2 t dx dt 33 2 t t 2 1

t 1 z 2tdt dzdt t 2 1

2

dt 4

3

t t 2 1

dt 4 arctanh t

3 2

dz z

4 arctanh t

dx x 2 pxq

p 2

dx x 2p 2

p2 4

q

dx x 2p 2

p2 4

q

dx x 2p 2

p 2 4q 4

3 ln z 4 arctanh t 3 lnt 2 1 4 arctanh t 2 2 ln x 2 2 1 4 arctanhx 2 3 lnx 1 3 lnx 3 4 arctanhx 2 2 2

x

t z

dx dtp 2 4q 4 x 2 3x4 x 2 3x4 2x3 9 5 x 3 22

1260

x1 2 x 2 3x4 5 2 5 2

dx

x x2

2x 6 5

x 2 2x1 x 2 3x4

dx 5 2

x 2 3x4 2x3 x 2 3x4

x 9 2

dx

x 2 3x42x13x4 x 2 3x4 2x 6 33 5 x 3 22 2

dx 5 2

dx

5x3 x 2 3x4

ln1 2

dt t 2 z

ln t t 2 z2

ln x

p 2

x

p 2

2

p 2 4q 4

9 4 4

x

p x x xp q3x6 x 2 4x5

C dx

7 4

1265

x 3 2 3 2

7 4

dx

x 3x 4 t x 2x 3dx dt x5 2

z5 2

3 2

2x4 x 2 4x5

dx dzdz z2 7 4

x 2 4x 5 t ln t 9 7

dt t

9 2

x

7 arctan

2 7

z 7

2x 4dx dt

1266

3 2

dt t

3 t 3 x 2 4x 5 Cx 1 4 1 2 25 4

arcsin 1270

2 5

1 x

1 2

arcsin

2 5

2x arcsin 2x

2x x 5

C

2x8 1xx 2

dx x 1 21 2 2

2x8 1 x 1 22

1 45 4

dx 2 dx x 1 21 2 2

x45 4

x 1 2

2

dx 2

x 1 2

2

dx

dx x1 x 2 2

25 4

5 4

x

dx 9 t xdz5 4

x1 t dx dt dt t t1 2 2 1 z

x 1 2 dt t

1 2

2

z2 5

dt t t 2 2t1

2x 1267

dx dtz 2

dx dz 2 t 9 arcsin 5 z 2 1 x x 2 9 arcsin2x1 5

t C 1 z 1 z2

9

dt dz z2dz z2

x 5x 2 2x1

dx

1 10

10x22 x 2 2 x 1 5 5 1 5

dx

1 10

10x2 5x 2 2x1

dx

1 5 5

dx x 1 52 1 25 1 5

2 1 z1 t

dz z12zz 2 z2

1 x1

dz 12zz 2

2

dz 2z1 2

arcsin2x x1

z1 2

arcsin arcsin 1271

1

5x 2 2x 1 t x 10x 2dx dt 1 10 1 10

z

dx dzdx x dz1 5 2

2 2x 2 x1

arcsin Cdx x1 x1 2 1

1

2

arcsin

1x1 x1

arcsin

2

dt t1

1 5 5 1 5 5

4 25

5 25

dx x1 x 2 2x

2t 2

4 z 2 25

t 5

arcsinh

5 2

z

5x 2 2x1 5

5 25

arcsinh

5x1 2

C

x1 t dx dtdt t t 2 1

1268

dx x 1x 2

1 x2 t 1 x2 t2 xdx tdt

t

1 z

dt dz z2 dz z2 1 z

xt

tdt x

dx

tdt x2t

dt 1t 2

arctanh t arctanh 1 x C

2

1 1 z 1 t

2

dz z

1 1 z 1 x1

2

C

dz z 1z 2 z2

dz 1z 2

arcsin z

arcsin 1272

arcsin

1269x x 2 x1

x 2 2x 5 dx x 2 2x 1 4 dx x 1 2 4 dx1 t

x1 t 1 t2 dt t2

x1 t dx dt dt t1 t2

dx dt t2

1 1 t 1 2

1 1 t

dt t1tt 2 t2

dt t 1 t

1tt 2

dt 1tt 2

dt 1 t 1 22

1 4

t 2 4 dt t 2 sinh z dt 2 cosh zdz 4 sinh 2 z 4 2 cosh zdz 4 sinh 2 z 1 cosh zdz 4 cosh 2 zdz

t

z arcsin2 5

dt dzdz5 4

z 2

z arcsin

2 5

t

1 2

4

1 2

1 cosh 2zdz 2 1 cosh 2zdz 2 dz 2 cosh 2zdz

2z u 2dz du 2z cosh udu 2z sinh u 2z sinh 2z t t 2 arcsinh 2 sinh2 arcsinh 2 2 arcsinh 1273x1 2

1276

sinh2 arcsinhx1 2

cos x sin 2 x6 sin x12

dx

cos x sin x3 2 3

dx

C

sin x 3 t cos xdx dtdt t 2 3

x x 2 dx x1 2

1 4

x

1 2

dx

2

1 3

arctan

t 3

1 3

arctan

sin x3 3

C

1277

t1 4 1 4

t 2 t 2

ex 1e x e 2x

dx

dx dt

1 4 1 4

1 41 4 1 4

t 2 dt t 2

ex t dt1 4

e dx dt t2 1 4

x

t 2

dt at b a1 4

c1 4

t 2

dt| |1 4

t2 1278

dt 1tt 2

dt 1 t 1 22

1 4

dt3 4

t 1 2

2

arcsinh

2t1 3

arcsinh

2e x 1 3

C

t 2 t 2

t2

2tatb 2 t 2

c1 4

t 2

t 2 a 1 t 2 tat b c 4 t 2 t 2 2a tb c 1 a 4 a 1 , b 0 2 .c 1 a 1 ....c 1 1 4 4 8 4 c 1 8 I 1 4 1 4 t 2 1 4 1 2

1 4 1 4

sin xdx cos 2 x4 cos x1

sin x cos x2 2 3

dx

cos x 2 t sin xdx dt 1 8 1 2 dt t 2 3

ln t t 2 3

ln cos x 2 cos 2 x 4 cos x 1

C

t2

1 8

dt1 4

t 2 1 2 2

1 4 1 8

t 1 4t 2 arcsin 2x

arcsin 2t

1279 Iln xdx x 14 ln xln 2 x ln xdx x ln x2 2 41 ln xdx x 5ln x2 2

x

1 4x

2x 1 x x 2

1 8

arcsin2x 1 C1 4 1 4

ln x udx x 9 4

1274

duudu 5u2 2

2 x x 2 dx 2 x 1 4 1 2 9 4 1 2

x x 2 dx

x

1 2

2

I

t1 4

u2 z du dz z2 5z 2

dx dt t 2 dt 1 2

t 9 4t 2 1 2

9 8 9 8

arcsin arcsin9 8

2 3 2 3

t x 1 2 2 3

dz

zdz 5z 2

2

dz 5z 2

x

9 4x 2 1 4

2

2

x1 3

I1 C

zdz 5z 2

5 z2 t 2zdz dt arcsinz 5

dt 2

t

t 5 z2

x 2x x

2x x xdx2 x 2 2 1

arcsin

1275

I 1 5 u 2 2 1 u 2 4u 1 ln 2 x 4 ln x I2 dz 5z 2

xdx x 4 4x 2 3

xdx x 4 4x 2 41

arcsin

u2 5

arcsinln x2 5

ln x2 5

x2 2 t 2xdx dt 1 2

I I 1 2I 2 1 ln 2 x 4 ln x 2 arcsin 1280 Idx xaxb

C

dt t 2 1

1 2

arctanh t

1 2

arctanhx 2 2 C

1 xaxb

A xa

B xb

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 1281 I1 x2x3 x 2 5x9 x 2 5x61 ba

41 a 5b 2c 91 12a 4b 3c 43 4b 3c c 43 4 b 3 3 12a 4b 3 43 4 b 12a 43 91 3 3 4 b 43 4 b 55 7 b 2 3 3 3 3 2 47c 3c b 7, c 5, a 4 dx dx dx I 4 x1 7 x3 5 x4 4 lnx 1 7 lnx 3 5 lnx 4 C 1284

xa 1 lnx ba

1 ab

xb

1 ab

1 ba

dx xa

1 ab

dx xb

a

lnx b Cdx x 2 5x6

5dx x2x3

3 5x 3 2 dx xx5x 2 2 5x4 x 3 5x 2 4x 3 5x 2 4x 2 5x 2 4x x 5

dx

5x 3 2 xx1x4

dx 5

x3 2 5 xx1x4

dx

dx dx 3 a x2

x 3

b x3

|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 1 12 x1 1 x2 1 x3 3 4 x1x2x3 1 12 lnx 1 1 lnx 2 1 lnx 3 3 4 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 1 b 1 , a 12 , c 1 3 4 1283

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 1 2 4a 3b 4 1 3b 2 3b 5 2 b 7 3 1 7 c 25 2 3 c 161 6 dx dx I 1 1 dx 7 x1 161 x4 1 ln x 7 lnx 1 x 2 2 3 6 3 7 1 I 5x I 1 5x 2 ln x 3 lnx 1 161 lnx 4 C 6 1285

25x 2 20x2 xx1x4 2

xx1x4 b a x1 x 2

dx 5 dx c x4

25x 2 20x2 xx1x4

|xx 1x 4

161 6

lnx 4

dx xx1 2

x1 t dt t2 2

1 t

dx dt t2

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

2x 2 41x91 dx x1x3x4 2x 2 41x91 a x1 x1x3x4 2

1 dt 1t

1 t

dt 2 t 1t t dt 1t 2

t 1t 2

dt 1 1t

t11 1t 2

dtx x1

ln1 t

ln x1 x

C

b x3

c x4

|x 1x 3x 4

12863 xx4 1 4x 3 4 dx 1 4x 4x 3 x dx 1 4 4 4 4x 3 x x4 x x4 dx 4 1 x2x12x1 dx 4 x4x 2 1 b c x4 a 2x1 2x1 |x2x 12x x x2x12x1

x 3 1 dx 4x 3 x x 1 4 4

dx 1

1 4

x4 4x 3 x

dx

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 9 2 b 1c 1 9 7 2 2 I 4 dx x 1287 I x 4 6x 12x 6 : x 6x 12x 8 x x 4 6x 3 12x 2 8x 6 8x I xdx 2 4x33 dx x2 2 x 4 6x 3 12x 2 6 x 3 6x 2 12x8 3 2

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 49 499 4

x 2 8x7 x2 2 x5 2 2

a x2 3

b x2 2 2

c x5

d x5 2 2

|x 2 2 x 5 2

dx 2x1 dx

7 2

9 2

2x1

dx 4 ln x dx

7 4

ln2x 1

ln2x 1 C

I 1290

30 343 dx

x2

27 49

dx

x2 2

30 343

dx

x5

8 49 dx

x5 2

27 49x2

30 343

lnx 2

8 49x5

30 343

ln

x 4 6x 3 12x 2 6 x2 3 3 2

68x x 3 6x 2 12x8

2x3 3 x 2 3x2

dx

x 2 3x 2 t 2x 3dx dt dx 22 dx x2 3

8

x2 dx 22 x2 2

x2 2

2

dx x2 3

4x838 x2 3 8 x2 x2 2

dx

x2 2

8 C

x2 x2 3

1291

dt t3

1 2t 2

1 2 2x 2 3x2

C

11 x2 2

1288 I5x 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

dx|

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 15 2d 2b 3a 2b 3d 5d 3a 55 a 3a 8a 25 15 a 5, d 0 6 2b. . . b 3 2 0 I 5x 6x92 dx x 5x3 x 2 2x3 dx 2 2x3 2x 2x3

5x 2 6x9 2 x 2 2x3 2

ax 2 2x32x2axb x 2 2x3 22

cxd x 2 2x3

|x 2x 3

2

2 2

1 ax a bx cx ab 0 c 0a 1b 1 I 1 dx x 2x1 dx ln x x I x I 1 x ln x 1 2

3 x 3 x1 dx x xx1 dx 3 x xx 2 1 bxc 1 a x x 2 1 |xx 2 xx 2 1 2 2

dx 1

dx xx 2 1

1 2

lnx 2 1

lnx 1 Cdx x 4 1

2

1292

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 1 b 1 2 2 I1 Ix 12931 2

x4 dx x 4 1 1 x 2 1x 2 1 3

x 4 11 dx x 4 1 axb cxd x 2 1 x 2 1 2

dx x 2 2

x I1

|x 1x 1

5x3 x 2 2x3

C

x 2 1 x 2 1 1 arctanh x 1 2 2

dx

1 2

dx

1 2

arctanh x

1 2

arctan x

arctan x C

1289

x 2 8x7 2 x 2 3x10

dx

x 2 8x7 x2 2 x5 2

dx

dx x 2 4x3x 2 4x5

dx x3x1x 2 4x5

x 12 x1 x2

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 3 1 1 1 1 7 1 1 4 3 0 11 2

4 1612 2 42 3 2 42 1 2 1 x3x1x 2 4x5 3 2

I I1 a x3

1 52

lnx 3

1 20

lnx 1

1 65

2x 15 2 x 2 4x5

dx

1 65

2x4 7 2 x 2 4x5

1 65

2x4 x 2 4x5

7 1 2 65

dx x 2 4x5

b x1

3

cxd x 2 4x5 2

|x 3x 1x 2 4x 5

x 2 4x 5 t 2x 4dx dt I1 I1 I :4 : 5 1294

x2 z dx dz

0 1 3 0 1 1

0 0 1 0 0 0 1 5 0 0 0 1 5 0 1

3 1

5 1 1 11 2 1 2

1 0 0

1 2 8 0 1 1 1 0 1 1 8 5 1 7 15 2 1 23 5

1 7 2 5 0 0 0

0 1 3

0 0 1

dt 7 1 dx 7 dz 1 1 65 ln t 130 z 2 1 t 2 65 65 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 20 lnx 1 65 lnx 4x 5 130 arctanx 52

2 C

4 0

4 0

5 15

0 10

0 2 7 0 2 0 1 2 1

0 2 7 0 2 0 1 0 1 2 0 1 0 0 1 1 0 0

1 3 5 11 2 1 2

1 5 0 0 0

0 1 1 8 5 1

1 1 0 0 1 7 15 215 2 1 2 1 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 1 1 0 0 0 15 b 1 ,c 3 I1 3

dx dx x1x 2 x1 x 3 1 bxc 1 a x1 x 2 x1 x1x 2 x1 2 2

|x 1x 2 x 1

0 2 7 0 2 0 1 0 1 0 0 1 0 0 1 7 15 2 03 26

0 2 7 0 0

1 0 0 0 0 0 1 1 1 0 1 2 0 1

1 1 0 0 0 2 1 0 0 1 2 1 1 1 0 2 0 1 2 0 0

1 5 0 1 2 8 5 0 0 0

1 1 1 0

0 0 0 1 5 0 0 0 3

0 2 0 0 1

7 7 1 2 1 7 15 215 2 1 2

0 2

2 8 5 0 1 2815 5

1 5

0 2 4 22 ,a 1 3 3 1 x 2 3 3

0 0 3 2

0 1 2

0 2 0 0

0 2 0 0

x1

dx

I I I I I 1295

24 3

1 0 0 I I

1 0 0 d1 52 1 52

0 1 2

0 0 01 52

1 3 1 3 1 3 1 3 1 3

lnx 1 lnx 1 lnx 1

dx 1 lnx 1 1 x 2x2 3 3 x 2 x1 x1 1 6 x 2x4 2 x1 dx 1 x 2x1 1 x 2 x1 2 x1 6 2 dx 1 lnx 2 x 1 1 2 6 2 x 1 1 1 2 4

0 2

lnx 1 lnx 1 1

1 6 1 6

lnx 2 x 1 lnx 2 x 1

1 2

dx x 1 22

3 4 2x1 3

3 3

arctan

C

26 3 ,c 1 20

1 , b 20 , a dx x3

dx x1

2 x 3 65 26 x 2 4x5

2 65

dx

dx x 4 1

dx 2 x 2 1 2x 2 11 x 2 2 x1

lnx 3

1 20

lnx 1

1 65

2x

3 26

65

x 2 4x5

dx

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 2 1 0 1 0 0 1 0 0 1 0 0 0 0 d I I I1 2

b d

0 1 2 1 02 2

1 2 1 0 1 2 0 14 2 4 2 4 2 2 4

0 1 1 02 2

0 0 1 : 2 ~ 1 1 :2 2

1 1 0 0 1 14 2

02 2

1 1 0 0 1 1 0 0 0

02 2

0 0 0 1 ~

2 0

1 2 1 1

2 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 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 gauss 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 2 0 0 0 1 0 0 0 0 0 1 0

x 2 1 x 3 1x 3 1 2

2

ax 2 bxc x 3 1

dx 2 exf x 3 1 3

|x 3 1x 3 1

0 0 0 1 0 0 0 ~ ~

0

0 2 0 1 0 1 2

2 1

2

0 1 0 1 14 2 4 2

0 0

~

1 0 0 1

0 0 0 2

1 1 0 0 1 0 0 0 0

1 1

0 0 1 0 2 0 2 1

1 1

2 1

e0 d 1 f 1 2 2 1 c 2 0 c 1 b 0 2 a 1 0 a 1 2 2 I I1 21 2

0 1 ,b 1 22 4

0

x2 1 2 x 3 1

,c 2 4

,a x 1 2 2 4

2 4

1 x2 1 2 2 x 3 1

x 2 1

1 2

x 2 1 x 3 1

1 2

x 2 1 x 3 1

x 1 2

x 2 2 x1

x 2 2 x1

1 4

2 x2 x 2 2 x1

1 4

2 x2 x 2 2 x1

I2 8

2 4 2 8 2

x 2 x 2 2 x1 2x 2 2 x2 2 x1

2 8

x 2 x 2 2 x1 x2 2 x1 2

2 8

2x2 2 x 2 2 x1

2x2 2 x 2 2 x1

I

I ln t 2 2

x1 dx 1 x 3 1 dx 2x11 dx 1 2 2 x 2 x1 x 2 x1 dx 2x1 2x11 2 x1 dx 2 x1 2 x1 dx x x x dt dx dx x 2x1 dx x 2 x1 2 2 x1 t x 1 1 1 2 4

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 1 22

2x 2 2

3 4

dz z

dx x 1 22

1 1 4

I lnx x 1 I lnx x 2 x2

x 2x1 t 2x 2 dx dt I 2 8

x 2x1 z 2x 2 dx dz 2 8 2 8 1 4

2 3 3 2 3 3

arctan

2x1 3 3

ln z

dx x 1 22

3 4 2 3 3

dt t

2 8 1 8 1 8 2 8

dz z

2 2 x1 dx x 2 2 x1 dx x2 2 2

x2

2 8

2 8

I ln 1297dx2 2 2

x 2 x1 x 2 x1

2x1 3 1 arctan lnx 2 3 2 3 2 3 2x1 arctan 3 arctan 2x1 3 3 3

x 1 C

arctan

2x1 3 3

I 1 2 ln t 8 I 1 2 ln t 8 I I 12962 8 2 8

2 ln z 2 ln z ln z 2 4

1 1 2

2 x1 dx x 2 2 x1 1 4 x

?????????

1 1 2

2 4

dx 2 1x 2

ln t

arctan 2 x 1 2 8

2 4

arctan2 4

2x1 arctan

x tan t dx 1 tan 2 tdt1tan 2 tdt 1tan 2 t 1 22

ln x 2 2 x 1 x 2 1 x 4 x 2 1x 2 1

ln x 2 2 x 1 x 2 1 x 6 1

arctan 2 x 1

dt 1tan 2 t 1 2

dt 1 1 2sin 2 t cos 2 t

dtcos 2 tsin 2 t cos 2 t

1 4

cos 2 tdt cos 2 tsin 2 t 1 2

cos 2 tdt 1 1 4

cos 2 tdt sin2 arctan x

dx x 4 x 2 1

dx

dx

x 2 1 x 3 1x 3 1

dx

1 cos 2t

dt

cos 2tdt

1 2

t

sin 2t

arctan x

1298 I3x5 x 2 2x2

I1 dx 3x5 x1 2 1

dx 3

I 3 I 3 I1 3 2

x11 x1 2 1 x1 x1 2 1

dx 5 arctanx 1

x dx x1 2 1 x1 3 x1 2 1

dx 3 arctanx 1 5 arctanx 3t t 2 1

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

dx x 2 x1 1 1 4 4

dx x 1 22

3 4

I1 I1 I2 I2 I2 I2

xdt t2 3 4

1 2

t2 3

dx dt t

arctandx x 1 22

2 3 3 4

t2

2 3

arctan

2 3

x

1 2

2 3

arctan

2x1 3

x1 t dx dt

dt

t2 1 u 2tdt du

3

du 2u

3 2

ln u

dx 2 x 2 x1

lnt 2 1 I 1 3 ln x 1 2 1 C 2 I 3 ln x 1 2 1 2 arctanx 1 C 2

xdt t2 3 4

1 2

dx dt2

1299

u t t2 3 4 t t2 3 4 t t2 3 42 2 2

1 t2 3 42

dt dv tv dt 3dt t2 3 43

dx 2 x1x 2 x1 a x1 2

1 2 2x1x x1 2

256t 3 4t 2 3

dt du256t 2 3 4t 2 3 dt 2 4t 2 3

bxc 2 x 2 x1

dxe x 2 x1

|x 1x 2 x 1

2

I2 I2 I2

64

t t2 3 4 dt t2 3 43 2

64 t t2 3 4

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 2 0 0 2 1 0 3 1 0 2 2 0 2 1 1 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 2x1x x1

4t 2 33 3 4t 2 32

dtdt t2 3 42

3

dt t2 3 43

I2 3

. . . . odusta san

1 0 0 Gaussova eliminacija 0 0 1 0 0 0

1 0 0

0 0 0 0 1 0

0 1 0 1 2 0 0 0 0 1 1 1

lnx 1 lnx 1

dx x 2 x1

xdx 2 x 2 x1 dx 2 x 2 x1

xdx x 2 x1

dx x 2 x1 1 1 4 4

dx 2 x 2 x1

I lnx 1 I 1 I 2