Preuzmite Integrali iz Matematike: rešeni zadaci i više Skripte u PDF od Matematika samo na Docsity! 1031 5a2x6dx 5a2 x6dx 5a2 x77 5 7 a 2x7 C 1032 6x2 8x 3dx 6 x2dx 8 xdx 3 dx 6 x33 8 x2 2 3x 2x 3 4x2 3x C 1033 xx ax bdx xx2 xb ax ab x3 x2b ax2 xabdx x3dx b x2dx a x2dx ab xdx x44 b x3 3 a x3 3 ab x2 2 14 x 4 13 bx 3 13 ax 3 12 abx 2 C 1034 a bx32dx a2 2abx3 b2x6dx a2x 2ab x44 b2 x6 7 a2x 12 abx 4 17 b 2x6 C 1035 2px dx 2px t| 2pdx dt t dt2p 1 2p t 1 2 dt 12p 2 3 t 3 2 13p t 3 2 13p 2px 3 2 23 2 x px C 1036 dx n x x 1n dx x 1 n n n 1n n n x n1 n n1 n n x n1 n n1 C 1037 nx 1nn dx nx u ndx du u 1nn dun 1n u 1n n du 1n u 1n n n n 1n n n n 1n u 1 n 1 n u 1n nx 1 n C 1038 a 23 x 23 3dx a2 3a 43 x 23 3a 23 x 43 x2 dx a2x 3a 43 x 5 3 5 3 3a 23 x 7 3 7 3 x33 a 2x 95 a 4 3 x 5 3 97 a 2 3 x 7 3 13 x 3 C 1039 x 1x x 1dx x 32 1 dx x 25 x 5 2 C 1040 x21x22 3 x2 dx x4x22 x 2 3 dx x 103 x 43 2x 23 dx 313 x 13 3 37 x 7 3 2 3x 13 313 x 13 3 37 x 7 3 6 3 x C 1041 x mxn 2 x dx x2m2xmnx2n x dx x2mx 12 2xmnx 12 x2nx 12 dx x2m 12 2xmn 12 x2n 12 dx x 2m 12 1 2m 12 1 2 x mn 12 1 mn 12 1 x 2n 12 1 2n 12 1 C x 2m 12 2m 12 2 x mn 12 mn 12 x 2n 12 2n 12 C 1042 a x 4 ax dx ax 12 a 2 a x x2dx ax 12 a2 4a 32 x 6ax 4 a x 32 a 1 2 x 12 a2 4a 32 x 6ax 4 a x 32 x2 a 12 1 x a2 4a 32 6 x a 4x a a 1 2 x 12 1 12 1 a2 4a 32 x 6a 23 x 3 2 4 x22 a x 5 2 5 2 a 1 2 2a2 x 4a 32 x 4ax 32 2x 2a 32 x 4ax 4 a x 32 2x2 2 5 a x 5 2 C 1043 dx x27 1 7 arctan 1 7 x C 1044 dx x210 1 10 arctanh 1 10 x C 1045 dx 4x2 arcsinh 12 x C 1046 dx 8x2 arcsinh 14 2 x C 1047 2x2 2x2 4x2 dx 2x 2 2x2 2x2 2x2 dx 1 2x2 dx 1 2x2 dx arcsin 12 2 x arcsinh 1 2 2 x C 1048*a tan2xdx sin2x cos2x dx 1cos2x cos2x dx 1 cos2x dx dx 1cosx sinx x C 1048*b tanh2xdx sinh2 cosh2 1cosh2x cosh2 dx cosh2 dx sinhxcoshx x tanhx x C 1049a cot2xdx cos2x sin2x dx 1sin2x sin2x dx dx sin2x dx cotx x C 1049b coth2xdx cosh2x sinh2x dx 1sinh2x sinh2x dx dx sinh2x dx cothx x C 1050 3xexdx 3exdx 3ex u 3xex1 ln3dx du 3ex du3xex1ln3 du 1ln3 1 1ln3 u 3ex 1ln3 C 1051** adxax a dxax a x u| dx du a duu a lnu a lna x C 1052** 2x32x1 dx 2x12 2x1 dx dx 2 2x1 dx x 2 2x1 dx 2x 1 u| 2dx du x duu x lnu x ln2x 1 C 1054 xdx abx 1 b bxdx abx 1 b bxaa abx dx 1 b bxaa abx dx 1 b dx a b dx abx x b a b dx abx a bx u| bdx du x b a b du b u xb a b2 duu xb a b2 lnu x b a b2 lna bx C 1055 axbx a x ba x dx a xdx x b x dx a xx dx b dx x a x x dx b dx x a dx a x dx b dx x ax a dxx b dx x ax b a dxx ax ba dxx x u| dx du ax ba du u ax ba 2 duu ax ba 2 lnu a x ba 2 lnx C 1056 x21 x1 x2 1 : x 1 x 1 2 x1 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 x x3 dx 7 dx x3 x299 x3 dx 5 x33 x3 dx 7 lnx 3 x 3dx 9 dxx3 5 dx 15 dx x3 7 lnx 3 x 2 2 3x 9 lnx 3 5x 15 lnx 3 7 lnx 3 12 x 2 2x lnx 3 C 1058 x4x21 x1 dx x4 x1 dx x2 x1 dx dx x1 x411 x1 dx x211 x1 dx lnx 1 x41x1 dx lnx 1 x21 x1 dx lnx 1 lnx 1 3 lnx 1 x 21x1x1 x1 dx x1x1 x1 dx 3 lnx 1 x 1dx x2 1x 1dx 3 lnx 1 x22 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 1059 a bxa 2dx a2dx 2 baxa dx b 2 xa2 dx a2x 2ab lnx a b2 dx xa2 x a t| dx dt a2x 2ab lnx a b2 dt t2 xa2 2ab lnx a b2t xa2 2ab lnx a b2 xa C 1060* x x12 dx x11 x12 dx dxx1 dx x12 lnx 1 dx x12 x 1 t| dx dt lnx 1 dt t2 lnx 1 1t lnx 1 1 x1 C 1061 bdy 1y b dy 1y 1 y t| dy dt b dt t 2b t 2b 1 y C 1062 a bx dx 2 3x t| 3dx dt 4t u| 4t ln4dt du 4t dt3 1 3 4tdt 13 4t du 4t ln4 1 3 ln4 du 1 3 ln4 u 4t 3 ln4 1 3 ln4 4 23x C 1089 et etdt 2 etet2 dt 2 sinh tdt 2cosh t C et etdt etdt etdt et et C 1090 e xa e xa 2dx e2 xa dx 2 dx dx e2 x a 2 xa t| 2dx a dt et a2 dt 2x a 2 e tdt a2 etdt 2x a 2 etdt 12 ae 2 xa 2x 12 ae 2 xa C 1091 axbx2 axbx dx a2x2axbxb2x axbx dx ax bx dx 2dx bxax dx a b x dx 2x ba xdx a b x ln a b 2x b a x ln ba C 1092 a2x1 ax dx a2x 12 xdx a 12 xdx a 32 xdx a 12 xdx au 23 du at2dt 3 2 x u| 12 x t| 3 2 dx du dx 2 dt 23 lna a u 2lna a t 2lna a 3 2 x 3 a 12 x C 1093 ex21xdx ex21xdx x2 1 t| 2xdx dt et dt2 1 2 etdt 1 2 e t 12 e x21 C 1094 x7x2dx x2 t| 2xdx dt 7t dt2 1 2 7tdt 1 2 ln7 7 t 12 ln7 7 x2 C 1095 e 1 x x2 dx 1 x t| dx x2 dt etdt et e 1x C 1096 5 x dx x x t| dx 2 x dt 5t2dt 2 5tdt 2ln5 5t 2 ln5 5 x C 1097 ex ex1 dx ex 1 t| exdx dt dtt ln t lnex 1 C 1098 ex a bex dx a bex t| bexdx dt t dtb 1 b t dt 23b t 3 2 23b a be x 3 2 C 1099 e xa 1 13 e xa dx e x a 1 t| 1 a e x a dx dt t 13 adt a t 13 dt 34 at 4 3 34 ae x a 1 43 C 1100* dx2x3 2x 3 t| 2x ln2dx dt dt 2x ln2 t 1 ln2 dt 2xt 1 ln2 dt t3t 1 ln2 dt t23t 1ln2 dt t 32 2 94 1ln2 1 9 4 arctanh t 3 2 9 4 23 ln2 arctanh 2 3 t 1 2 3 ln2 arctanh 2 3 2 x 3 1 C 1101 axdx 1a2x ax u ax lnadx du du lna1u2 1lna du 1u2 1lna arctanu 1 lna arctana x C 1102 ebx 1e2bx dx ebx u bebxdx du du b 1u2 1 b du 1u2 1 b arctanhu 1 b arctanhebx C 1103 et 1e2t dt et u etdt du du 1u2 arcsinu arcsinet C 1104 sina bxdx a bx z bdx dz sin z dz b 1 b sin zdz 1 b cos z 1 b cosa bx C 1105 cos x 2 dx x 2 z dx 2 dz cos z 2 dz 2 cos zdz 2 sin z 2 sin x 2 C 1106 cosax sinax2dx sin2ax 2sinaxcosax cos2axdx 1 2sinaxcosaxdx 1 sin2axdx x sin2axdx 2ax z 2adx dz x sin z dz2a x 1 2a sin zdz x 1 2a cos z x 1 2a cos2ax C 1107 cos x dx x x z dx 2 x dz cos z2dz 2 cos zdz 2sin z 2sin x C 1108 sinlgx dxx sin lnxln2 dx x lnx ln2 z| dx x ln2 dz sinz ln2dz ln2 sin zdz ln2cos z ln2cos lnxln2 ln2cos lgx C 1109 sin2xdx 1cos2x2 1 2 dx 1 2 cos2xdx 2x z| 2dx dz x2 1 2 cos z 1 2 dz x 2 1 4 cos zdz 1 2 x 1 4 sin z 1 2 x 1 4 sin2x C 1110* cos2xdx 12 1 cos2x 2dx 1 2 dx 1 2 cos2xdx 1 2 x 1 2 cos2xdx 2x z| 2dx dz 12 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 sin2x C 1111 sec2ax bdx dx cos2axb ax b t adx dt dt a cos2t 1a dtcos2t 1 a tan t 1a tanax b C 1112 cot2axdx cos2ax sin2ax dx 1sin2ax sin2ax dx dx sin2ax dx ax t adx dt dt a sin2t x 1a dtsin2t x 1 a cot t x 1a cotax x C 1113 dxsin xa x a t dx a dt adtsin t a dtsin t a ln tan t2 a ln tan x a 2 a lntan x 2a C 1114 dx3cos5x 4 5x 4 t 5dx dt dt 5 3cos t 1 15 dt cos t 1 15 ln tan t 2 4 115 lntan 1 2 5x 4 1 4 1 15 ln tan 5 2 x 1 8 C 1115 dxsinaxb ax b t adx dt dt a sin t 1 a dtsin t 1a ln tan t2 1 a ln tan axb2 C 1116 xdx cos2x2 x2 t 2xdx dt dt 2 cos2t 12 dt cos2t 12 tan t 1 2 tanx 2 C 1117 x sin1 x2dx 1 x2 t 2xdx dt sin t dt2 1 2 sin tdt 1 2 cos t 1 2 cos1 x 2 C 1118 1 sinx 2 1 2 dx 1 sin2x 2 2 sinx 2 1 dx x 2 dx sinx 2 dx sin2x 2 x 2 t 2 dx dt x 2 dt 2 sin t dt 2 sin2t x 2 dtsin t 2 2 dt sin2t x 2 ln tan t2 2 2 cot t x 2 ln tan x 22 2 2 cotx 2 C 1119 tanxdx sinxcosx dx 1cos2x 1cos2x dx 1cos2x 1cos2x 1cos2x 1cos2x dx 1cos 22x 1cos2x dx sin22x 1cos2x dx sin2x 1cos2x dx 1 cos2x t| 2sin2xdx dt dt 2 t dx 1 2 ln t 1 2 ln1 cos2x C 1120 cotxdx cosxsinx dx 1 2 1cos2x 1 2 1cos2x dx 1cos2x 1cos2x dx 1cos2x 1cos2x 1cos2x 1cos2x dx 1cos 22x 1cos2x sin2x1cos2x dx 1 cos2x t| 2sin2xdx dt dt 2 t 1 2 ln t 1 2 ln1 cos2x C 1121 cot x ab dx x ab u| dx ab du cotua bdu a b cotudu a b lnsinu a b ln sin xab C 1122 dxtan x5 cot x 5 dx x 5 u| dx 5 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| dx 2 x du 2 tanudu 2 tanu 2 sinucosu du cothxdx coshxsinhx dx sinhx t coshxdx dt dtt ln t ln sinhx C 1145 x 5 5 x2 dx 5 x2 t 2xdx dt 5 t dt2 1 2 5 t dt 5 12 t 6 5 512 5 x 2 6 5 C 1146 x31 x44x1 dx x4 4x 1 t 4x3 4dx dt dt 4 t 1 4 ln t 1 4 lnx 4 4x 1 C 1147 x3 x85 dx x4 t 4x3dx dt dt 4 t25 14 dt t25 520 arctan t 5 5 5 20 arctan x4 5 5 C 1148 xex2dx x2 u 2xdx du eu2 du 1 2 eudu 1 2 e u 12 e x2 C 1149 3 23x 2 23x2 dx 2 3x2 t2 3xdx tdt 3t t2 tdt 3x 1 3 3t t2 t 22 3 tdt 33 3t t t22 dt 33 3 t t22 33 t t t22 3 1 t t22 t2 2 z2 tdt zdz 13 3 ln t t 2 2 3 1tz zdzt 13 3 ln t t2 2 3 1 z22z zdz 13 3 ln t t 2 2 3 1 z22 dz 13 3 ln t t 2 2 12 3 2 arctan 1 2 13 3 ln t t 2 2 12 3 2 arctan 2 2 t 2 2 13 3 ln 2 3x 2 2 3x2 2 12 3 2 arctan 2 2 2 3x 2 2 13 3 ln 2 3x 2 3 x 12 3 2 arctan 1 2 2 3 x C 3 23x 2 23x2 dx 3 23x2 dx dx 23x2 dx x2 23 33 dx x2 23 12 6 arctan 1 2 x 6 3 3 arcsinh 1 2 x 6 1 2 6 arctan 1 2 x 6 1 3 3 arcsinh 1 2 x 6 C 1150 x 31 x1 dx x 2dx xdx dx 2 dxx1 x3 1 : x 1 x2 x 1 2x1 x3 x2 1 x2 x2 x 1 x x 1 2 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 C 1152 1sinxxcosx dx x cosx u 1 sinxdx du duu lnu lnx cosx C 1153 tan3xcot3xsin3x dx 3x u 3dx du 13 tanucotu sinu du 1 3 sinu cosu cosu sinu sinu du 1 3 1 cosu 13 cosu sin2u sinu t cosudu dt 13 ln 1sinu cosu 13 dt t2 13 ln 1sinu cosu 13t 1 3 ln 1sinu cosu 13sinu 1 3 ln 1sin3x cos3x 1 3sin3x C 1154 dx x ln2x lnx t dx x dt dt t2 1t 1 lnx C 1155 sec2x tan2x2 dx dx cos2x tan2x2 tanx t dx cos2x dt dt t22 ln t t2 2 ln tanx tan2x 2 C 1156 2 x 2x21 dx 2x21 2 2x21 dx x 2x212 dx 2x2 1 t 4xdx dt dx x2 12 dt 4t2 2 arctanx 2 14t 2 arctanx 2 1 42x21 C 1157 asinx cosxdx sinx t cosxdx dt atdt 1lna at 1 lna a sinx C 1158 x2 3 x31 dx x3 1 u| 3x2dx du du3 3 u 1 2 3 u2 12 3 x 3 12 C 1159 xdx 1x4 x2 u| 2xdx du du 2 1u2 12 arcsinu 1 2 arcsinx 2 C 1160 tan2axdx ax u| adx du 1a tan2udu 1a sin 2u cos2u du 1a 1cos 2u cos2u du 1a ducos2u 1 a du 1a tanu u2 1 a tanax 12 ax C 1161 sin2 x2 dx x 2 t dx 2 dt 2 sin2tdt 2 12 1 cos2t 1 cos2t t 1 2 sin2t x 2 1 2 sinx C 1162 sec2xdx 4tan2x tanx t dx cos2x dt dt 4t2 arcsin t2 arcsin tanx 2 C 1163 dxcos xa x a t dx a dt a dtcos t a lnsec t tan t a ln 1sin xa cos xa C 1164 3 1lnxx dx 1 lnx t dx x dt 3 t dt 34 3 t4 3 4 3 1 lnx4 C 1165 tan x 1 dx x1 x 1 t dx 2 x1 dt 2 tan tdt 2 lncos t 2 ln cos x 1 C 1166 xdx sinx2 x2 t 2xdx dt 12 dt sin t 1 2 ln 1cos t sin t 1 2 ln 1cosx2 sinx2 C 1167 e arctanxx ln1x2 1 1x2 dx earctanx 1x2 dx x ln1x 2 1x2 dx dx 1x2 arctanx u dx 1x2 du ln1 x2 t 2xdx 1x2 dt eudu 12 tdt arctanx eu 1 4 t 2 arctanx earctanx 14 ln 21 x2 arctanx C 1168 sinxcosxsinxcosx dx sinx 2t 1t2 cosx 1t2 1t2 dx 2dt 1t2 2t 1t2 1t2 1t2 2t 1t2 1t 2 1t2 2dt 1t2 2 t22t1 2t1t2 1t2 dt t22t1 2t1t2 1t2 atb 2t1t2 ctd 1t2 t2 2t 1 at b1 t2 ct d2t 1 t2 t2 2t 1 at at3 b bt2 2ct2 ct ct3 2dt d dt2 t2 2t 1 t3a c t2b 2c d ta c 2d b d a c 0 b 2c d 1 a c 2d 2 b d 1 a 1,d 0,c 1,b 1 t22t1 2t1t2 1t2 dt t1 2t1t2 dt t 1t2 dt 2t 1 t2 z 21 tdt dz 1 t2 u 2tdt du 2 dz2z du 2u 2 2 ln z 2 2 lnu ln2t 1 t2 ln1 t2 ln2 tan x2 1 tan 2 x 2 ln1 tan 2 x 2 C sinxcosxsinxcosx dx sinxcosxsinxcosx sinxcosxsinxcosx dx sin 2x2sinxcosxcos2x sin2xcos2x dx 12sinxcosx sin2xcos2x dx 1sin2xcos2x dx 1sin2x cos2x dx dx cos2x tan2xdx 2x t 2dx dt 12 dt cos t 1 2 tan tdt 1 2 ln 1sin t cos t 1 2 lncos t 12 ln 1sin2x cos2x 1 2 lncos2x C 1169 1sin x 2 2 sin x 2 dx x 2 t dx 2 dt 2 1sin t 2 sin t 2 1sin t2 sin t 2 12sin tsin 2t sin t dt 2 dtsin t 2 dt sin t 2 ln 1cos tsin t 2t cos t 2 ln 1cos x 2 sin x 2 2 x 2 cos x 2 C 1170 x2 x22 dx x222 x22 dx dx 2 dx x22 x 2 arctanh 12 x 2 C 1171 1x 2 x1x2 dx x22x1 x1x2 dx x 1x2 dx 2 dx 1x2 dx x1x2 x 1x2 dx 2arctanx dx x1x2 1 x2 t 2xdx dt dt 2 t 2arctanx dxx1x2 1 2 ln t 2arctanx dx x1x2 1 x1x2 ax bxc1x2 1 a1 x2 xbx c a ax2 bx2 xc 1 x2a b xc a a b 0 a 1b 1c 0 I1 ax dx bxc1x2 dx a lnx x 1x2 dx lnx 12 ln1 x 2 I 12 ln1 x 2 2arctanx lnx 12 ln1 x 2 2arctanx lnx C 1172 x2dt x 1t 22 1 t dt 2t2 t2 dt 2t2 arcsin 22 t arcsin 2 2x C 1191b dx ex1 x ln t dx dtt dt t eln 1 t 1 dt t 1 t 1 dt t 1t 1 dt1t ln1 t ln1 1 ex C 1191c x5x2 37dx 5x2 3 t 10xdx dt x5x2 37 dt10x 1 10 t7dt 1 80 t 8 180 5x 2 38 C 1191d xdx x1 x 1 t dx 2 x1 dt 2 xdt 2 t2 1dt 23 t3 2t 2 3 x 1 3 2 x 1 C 1191e cosxdx 1sin2x t sinx dt cosxdx dt 1t2 arcsinh t arcsinh sinx C 1192 x2x 510dx 2x 5 t x t52 2dx dt t52 t10 dt 2 1 4 t 5t10dt 1 48 t 12 544 t 11 148 2x 5 12 544 2x 5 11 C 1193 1x 1 x dx x t dx 2 x dt 1x1 x 2 x dt 2 1t2 1t tdt 2 tt3 1t dt t3 t : t 1 t2 t 2 2 t1 t3 t2 t2 t t2 t 2t 2t 2 2 2 t2dt tdt 2 dt 2 dtt1 2 3 t 3 t2 4t 4 ln1 t 23 x 3 x 2 4 x 4 ln1 x C 1194 dx x 2x1 2x 1 t 2x 1 t2 dx 2x1 dt x t 21 2 2x1 x 2x1 dt dtx dtt21 2 2 dt t21 2arctanh t 2arctanh 2x 1 C 1195 dx ex1 ex 1 t exdx 2 ex1 dt 2 ex1 ex dt ex1 2 dtex 2 dt t21 2arctan t 2arctan ex 1 C 1196 ln2xdx x ln4x ln2xdx xln2xln2 ln2x t dx x t txdt xtln2 tdt tln2 tln2ln2 tln2 dt dt ln2 dt tln2 t ln2 lnt ln2 ln2x ln2 lnln2x ln2 C 1197 arcsinx 2 1x2 dx arcsinx t dx 1x2 dt t2dt 13 t3 1 3 arcsin 3x C 1198 e2x ex1 ex 1 t exdx 2 ex1 dt e2xt 2 ex1 ex dt 2 t21t t dt 2 t2 1dt 23 t3 2t 2 3 e x 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 2 5 cosx 5 2 cosx C 1200* dx x 1x2 1 x2 t|2 1 x2 t2 xdx tdt tdt x xt 1x2 dt 1 t21 dt arctanh t arctanh 1 x2 C 1201 x2 1x2 dx x cos t dx sin tdt cos2t 1cos2t sin tdt cos2t sin tsin t dt cos2tdt 1cos2t2 1 2 t 1 4 sin2t 12 arccosx 1 4 sin2arccosx C 1202 x3 2x2 dx 2 x2 cos t 2xdx sin tdt x3 cos t sin tdt 2x x2 cos t sin tdt 2 2cos t cos t sin tdt 2 2sin tcos t sin t 2 cos t dt sin t cos t dt 12 cos t sin tdt cos t u sin tdt du du u 12 u du 2 u 1 3 u 3 2 2 cos t 13 cos t 3 2 2 2 x2 13 2 x 2 3 2 C 1203 x2a2x dx x2 a2 sin2t xdx sin tcos tdt sin tx sin tcos tdtx sin 2tcos tdt x2 sin2tcos tdt sin2ta2 sin t z cos tdt dz z2dz z2a2 dz a2 dz z2a2 z aarctan za sin t aarctan sin ta x2 a2 aarctan x 2a2 a C 1204* dx x x21 x 1t dx dt t2 dt t2 1 t 1 t2 1 dt t2 1 t 1t2 t2 dt t 1t2 t2 dt t 1 t 1t 2 dt 1t2 arcsin t arcsin 1x C 1205 x21x dx x sinh t dx cosh t sinh 2t1 sinh t cosh t sinh2t1 sinh t cosh t cosh2t sinh t sinh2t1 sinh t sinh t dt sinh t cosh t 2arctanhet cosh arcsinhx 2arctanhearcsinhx C 1206* dx x2 4x2 x 2sin t dx 2cos tdt 2cos tdt 2sin t2 42sin t2 2cos t 4sin2t 44sin2t dt 2cos t 4sin2t 44sin2t dt 2cos t 8sin2t 1sin2t dt 2cos t 8sin2t cos2t 2cos t 8sin2tcos t dt 14 dt sin2t 14 tan t 1 4 tanarcsin x 2 C dx x2 4x2 x 1t dx dt t2 dt t2 1 t2 4 1 t2 dt 4 1 t2 dt 1 t 4t 21 12 t t2 14 dt 12 1 2 4t 2 1 14 4t 2 1 14 4 1 x 2 1 14x 4 x 2 C 1207 1 x2 dx 1x2 1x2 dx 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 c 1 x2 a 2ax2 xb c 1 x2 x22a xb a c a 12 ,b 0 a c 1. . . .c 12 I 12 x 1 x 2 12 dx 1x2 12 x 1 x 2 12 arcsinx C 1208 dx x1x x sin2t dx 2sin tcos tdt 2sin tcos t sin2t1sin2t dt 2 sin tcos t sin t 1sin2t dt 2 sin tcos t sin t cos2t dt 2 sin tcos tsin tcos t dt 2 dt 2t 2arcsin x C 1209 a2 x2 dx x a sinh t dx 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 14 a 2 sinh2t 12 a 2t 14 a 2 sinh2arcsinh xa 12 a 2 arcsinh xa 12 x x 2 a2 12 a 2 arcsinh xa C 1210 x2 x2a2 dx x acosh t dx a sinh tdt a2 cosh2t a2 cosh2ta2 dt a2 cosh2t a cosh2t1 dt a cosh2t cosh2t1 dt a 1sinh2tsinh t dt a dt sinh t a sinh tdt acosh t a ln tanh t 2 acosharccosh xa a ln tanh arccosh xa 2 x a lntanh 1 2 arccosh x a C 1211 lnxdx dx dv lnx u x v dxx du x lnx x dxx x lnx x C 1212 arctanxdx arctanx u dx dv dx 1x2 du x v xarctanx x 1x2 dx 1 x2 t 2xdx dt xarctanx dt 2 t xarctanx 1 2 ln t xarctanx 1 2 ln1 x 2 C 1213 arcsinxdx arcsinx u dx dv dx 1x2 du x v xarcsinx x 1x2 dx 1 x2 t 2xdx dt xarcsinx dt 2 t xarcsinx 12 dt t xarcsinx t xarcsinx 1 x2 C 1214 x sinxdx xdx dv arcsinx u 1 2 x 2 v dx 1x2 du 12 x 2 arcsinx 12 x2 dx 1x2 12 x 2 arcsinx 12 1x21 1x2 dx 12 x 2 arcsinx 12 1 x2 dx 1 2 dx 1x2 12 x 2 arcsinx 14 x 1 x 2 14 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 1x2 dx x ln x 1 x2 x 1x2 dx 1 x2 t 2xdx dt x ln x 1 x2 dt 2 t x ln x 1 x2 t x ln x 1 x2 1 x2 C 1230 xdx sin2x x u dx sin2x dv dx du cotx v xcotx cotxdx xcotx cotxdx xcotx lnsinx C 1231 I xcosx sin2x cosx sin2x dx dv x u 1sinx v dx du I1 cosxsin2x sinx t cosxdx dt dt t2 1t 1 sinx I x 1sinx 1sinx dx xsinx dxsinx xsinx ln tan x2 C 1232 ex sinxdx exdx dv sinx u ex v cosxdx du I ex sinx ex cosxdx I1 ex cosxdx exdx dv cosx u ex v sinxdx du ex cosx ex sinxdx I1 ex cosx ex sinxdx I ex sinx ex cosx ex sinxdx ex sinx ex cosx I 2I ex sinx ex cosx I e x sinxex cosx 2 C ex sinxdx ex u sinxdx dv exdx du cosx v ex cosx cosxexdx ex cosx cosxexdx ex u cosxdx dv exdx du sinx v I ex cosx ex sinx sinxexdx I ex cosx ex sinx I 2I ex cosx ex sinx I e x cosxex sinx 2 C 1233 3x cosxdx 3xdx dv cosx u 3x ln3 v sinx du 3 x ln3 cosx sinx 3x ln3 dx I 3 x ln3 cosx 1 ln3 3x sinxdx 3xdx dv sinx u 3x ln3 v cosx du I 3 x ln3 cosx 1 ln3 sinx 3x ln3 3x ln3 cosx 3x ln3 cosx 3x ln23 sinx 1 ln23 3x cosx I 3 x ln3 cosx 3x ln23 sinx 1 ln23 I I 1 1 ln23 3 x ln3 cosx 3x ln23 sinx I 3x ln3 cosx 3x ln23 sinx ln231 ln23 3x ln3 cosx 3x ln23 sinx ln231 ln23 3x cosx ln3sinx ln231 C 1234 eax sinbxdx bx t bdx dt e atb sin t dt b 1 b e atb sin tdt e at b u sin tdt dv a b ea t b dt du cos t v 1 b e at b cos t a b ea tb cos tdt e at b u cos tdt dv a b ea t b dt du sin t v I 1 b e at b cos t a b2 e at b sin t a 1 b ea tb sin tdt I 1 b e at b cos t a b2 e at b sin t aI I 1 b e at b cos t a b2 e at b sin t a2 b2 I I 1 a2 b2 1 b e at b cos t a b2 e at b sin t I 1 b e at b cos t a b2 e at b sin t a2b2 b2 ea t b bcos ta sin t a2b2 ea bx b bcosbxa sinbx a2b2 I bcosbxa sinbx a2b2 eax C 1235 sinlnxdx lnx t dx x dt x sin tdt et sin tdt etdt dv sin t u et v cos tdt du I et sin t et cos tdt I1 et cos tdt etdt dv cos t u et v sin tdt du I1 et cos t et sin tdt et cos t et sin tdt I et sin t et cos t et sin tdt et sin t et cos t I 2I et sin t et cos t I e t sin tet cos t 2 elnx sin lnxelnx cos lnx 2 1 2 x sinlnx 1 2 xcoslnx C 1236 x3ex2dx x2 t 2xdx dt x3et dt2x 1 2 x2etdt 1 2 tetdt t u etdt dv dt du et v 12 te t etdt 12 tet 1 2 e t 12 x 2ex2 12 e x2 12 x 2ex 2 12 e x2 C 1237 e x dx x t|2 x t2 dx 2tdt 2 ettdt t u etdt dv dt du et v 2 tet etdt 2tet 2et 2 x e x 2e x C 1238 x2 2x 3 lnxdx x2 2x 3dx dv lnx u x3 3 x 2 3x v dxx du x 3 3 x 2 3x lnx x33 x2 3x dx x x 3 3 x 2 3x lnx x23 x 3 dx x 3 3 x 2 3x lnx x39 x2 2 3x C 1239 x ln 1x1x dx ln 1x1x u dv xdx 2 x21 dx du v x 2 2 x 2 2 ln 1x 1x x2 2 2 x21 x 2 2 ln 1x 1x x2 x21 x 2 2 ln 1x 1x x211 x21 x 2 2 ln 1x 1x dx dx x21 12 x 2 ln 1x1x x arctanhx C 1240 ln2x x2 dx ln2x u dv dx x2 2 lnx x dx du v 1x 1x ln2x 1x 2 lnxx dx 1x ln2x 2 lnxx2 dx lnx u dv dx x2 dx x du v 1x 1x ln2x 2 1x lnx 1x dxx 1x ln2x 2 1x lnx dxx2 1 x ln2x 2x lnx 2x C 1241 lnlnxx dx lnlnx u dxx dv dx x lnx du lnx v lnx lnlnx lnx dxx lnx lnx lnlnx dx x lnx lnlnx lnx C 1242 x2 arctan3xdx 3x t 3dx dt t29 arctan t dt 3 1 27 t2 arctan tdt arctan t u t2dt dv dt 1t2 du t 3 3 v 127 t3 3 arctan t t3 3 dt 1t2 127 t3 3 arctan t 1 3 t3 1t2 dt 181 t 3 arctan t t3 1t2 dt 1 t2 z 2tdt dz 181 t 3 arctan t t3z dz2t 181 t 3 arctan t 12 t2 z dz 181 t 3 arctan t 12 z1 z dz 181 t 3 arctan t 12 dz 1 2 dz z 181 t 3 arctan t 12 z 1 2 ln z 181 t 3 arctan t 12 1 t 2 12 ln1 t 2 181 3x 3 arctan3x 12 1 3x 2 12 ln 1 3x 2 181 27x 3 arctan3x 12 9 2 x 2 12 ln1 9x 2 13 x 3 arctan3x 1162 1 18 x 2 1162 ln1 9x 2 C 1243 xarctan2xdx arctan2x u xdx dv 2arctanx 1x2 dx du x 2 2 v x 2 2 arctan 2x x22 2arctanx 1x2 x 2 2 arctan 2x x2 arctanx 1x2 arctanx u x2 1x2 dx dv dx 1x2 du x211 1x2 dx x arctanx v x 2 2 arctan 2x x arctanxarctanx x arctanx dx 1x2 x 2 2 arctan 2x xarctanx arctan2x x 1x2 dx arctanx 1x2 dx x 2 2 arctan 2x xarctanx arctan2x I1 I2 I1 x1x2 dx 1 x2 t 2xdx dt 12 dt t 1 2 ln t 1 2 ln1 x 2 I2 arctanx1x2 dx arctanx t dx 1x2 dt tdt 12 t2 1 2 arctan 2x x 2 2 arctan 2x xarctanx arctan2x 12 ln1 x 2 12 arctan 2x 12 x 2 arctan2x xarctanx 12 arctan 2x 12 ln1 x 2 C 1244 I arcsinx2dx dv dx arcsin2x u v x 2 arcsinx 1x2 dx du I xarcsin2x x2 arcsinx 1x2 dx Iv x arcsinx 1x2 arcsinx u x 1x2 dx dv 1 1x2 dx du 1 x2 v Iv uv vdu 1 x2 arcsinx 1 x2 1 1x2 dx 1 x2 arcsinx x Iv x 1x2 1 x2 t 2xdx dt I 92 arcsin x 3 x 2 9 x 2 C 1255 dx x22x5 dx x22x14 dx x124 12 arctan x1 2 C 1256 dx x22x dx x22x11 dx x121 arctanhx 1 C 1257 dx 3x2x1 13 dx x2 13 x 1 3 13 dx x 16 2 136 1 3 13 dx x 16 2 1136 2 11 arctan 6x1 11 C 1258 xdx x27x13 xdx x 72 2 494 13 x 7 2 x 72 2 34 dx 72 dx x 72 2 34 x 72 t dx dt t t2 34 dt 72 dx x 72 2 34 12 ln4t 2 3 73 3 arctan 1 3 2x 7 3 12 ln 4x 7 2 2 3 7 3 arctan 2x7 3 ln2 12 lnx 2 7x 13 7 3 arctan 2x7 3 C 1259 3x2 x24x3 dx 3x2 x2243 dx 3x2 x221 dx 3 xdx x221 2 dx x221 3 x22 x221 dx 2 dx x221 3 x2 x221 dx 6 dx x221 2 dx x221 3 x2 x221 dx 4 dx x221 x 2 t dx dt t2 1 z 2tdt dz 3 t t21 dt 4 dt t21 3 t t21 dt 4arctanh t 32 dz z 4arctanh t 32 ln z 4arctanh t 3 2 lnt 2 1 4arctanh t 3 2 ln x 2 2 1 4arctanhx 2 32 lnx 1 3 2 lnx 3 4arctanhx 2 1260 x1 2 x23x4 dx x22x1 x23x4 dx x23x42x13x4 x23x4 dx x23x4 x23x4 dx 5x3 x23x4 x 52 2x 65 x23x4 x 52 2x 65 33 x 32 2 94 4 x 52 2x3 95 x 32 2 74 x 52 2x3 x23x4 92 dx x 32 2 74 x2 3x 4 t x 32 z 2x 3dx dt dx dz x 52 dt t 9 2 dz z2 74 x 52 ln t 9 7 7 arctan 2 7 z 7 x 52 lnx 2 3x 4 9 7 arctan 2x3 7 C 1261 x2 x26x10 dx x26x106x10 x26x10 dx dx 6x10 x32910 dx x 3 2x6 10 3 6 x321 dx x 3 2x6 x26x10 dx 8 dx x321 x2 6x 10 t x 3 z 2x 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 3 2 3 2 xx 2 1 2 dx 33 16 x 3 4 2 x 34 t dx dt 1 2 dt 33 16 t 2 1 2 arcsin 4t 33 1 2 arcsin 4x3 33 C 1263 dx xx2 dx 1 4 x 1 2 2 x 12 t dx dt dt 1 4 t 2 arcsin2t arcsin2x 1 C 1264 dx x2pxq dx x p2 2 p 2 4 q dx x p2 2 p 2 4 q dx x p2 2 p 24q 4 x p2 t dx dt p24q 4 z dt t2z ln t t2 z ln x p2 x p 2 2 p 24q 4 ln 12 p x x 2 xp q C 1265 3x6 x24x5 dx 32 2x4 x24x5 dx x2 4x 5 t 2x 4dx dt 32 dt t 3 t 3 x2 4x 5 C 1266 2x8 1xx2 dx 2x8 1 x 12 2 14 dx 2 x4 5 4 x 1 2 2 dx 2 x 1 2 4 1 2 5 4 x 1 2 2 dx 2 x 1 2 5 4 x 1 2 2 dx 9 dx 5 4 x 1 2 2 5 4 x 1 2 2 t x 12 z 2x 12 dx dt dx dz dt t 9 dz 5 4 z 2 2 t 9arcsin 25 5 z 2 1 x x 2 9arcsin 2x1 5 C 1267 x 5x22x1 dx 110 10x22 x2 25 x 1 5 dx 110 10x2 5x22x1 dx 1 5 5 dx x 15 2 125 1 5 5x2 2x 1 t x 15 z 10x 2dx dt dx dz 110 dt t 1 5 5 dx x 15 2 425 110 2t 1 2 1 5 5 dz z2 425 t5 5 25 arcsinh 5 2 z 5x22x1 5 5 25 arcsinh 5x1 2 C 1268 dx x 1x2 1 x2 t 1 x2 t2 xdx tdt tdt x xt tdtx2t dt 1t2 arctanh t arctanh 1 x2 C 1269 dx x x2x1 x 1t dx dt t2 dt t2 1 t 1 t2 1t 1 dt t 1 t2 1t 1 dt t 1tt2 t2 dt t 1 t 1tt 2 dt 1tt2 dt 1 t 12 2 14 t 12 z dt dz dz 5 4 z 2 arcsin 2 5 z arcsin 2 5 t 12 arcsin 2 5 1x 12 arcsin 2 5 2x2x arcsin 2x x 5 C 1270 dx x1 x22 x 1 t dx dt dt t t122 dt t t22t1 t 1z dt dz z2 dz z2 1 z 1 z2 2z 1 dz z 12zz2 z2 dz 12zz2 dz 2z12 arcsin z1 2 arcsin 1 t 1 2 arcsin 1 x1 1 2 arcsin 1x1 x1 2 arcsin 2x x1 2 arcsin 2x 2 x1 C 1271 dx x1 x22x dx x1 x121 x 1 t dx dt dt t t21 t 1z dt dz z2 dz z2 1 z 1 z 21 dz z 1z 21 dz z 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 t dx dt t2 4 dt t 2sinh z dt 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 u 2dz du 2z coshudu 2z sinhu 2z sinh2z 2arcsinh t2 sinh2arcsinh t 2 2arcsinh x1 2 sinh2arcsinh x1 2 C 1273 x x2 dx 14 x 1 2 2 dx x 12 t dx dt 14 t2 dt 1 4 t 2 1 4 t 2 dt 1 4 t 2 1 4 t 2 dt at b 14 t 2 c 1 4 t 2 dt| 1 4 t 2 1 4 t 2 a 14 t 2 2tatb 2 14 t 2 c 1 4 t 2 | 14 t 2 1 4 t 2 a 14 t 2 tat b c 1 4 t 2 t22a tb c 14 a a 12 ,b 0 .c 14 a 1 4 . . . .c 1 8 1 4 c 18 I t2 1 4 t 2 18 dt 1 4 t 2 14 t 1 4t 2 18 arcsin2t 14 x 1 2 1 4x 1 2 2 18 arcsin2x 1 2 14 2x 1 x x 2 18 arcsin2x 1 C 1274 2 x x2 dx 2 14 1 4 x x 2 dx 94 x 1 2 2 x 12 t dx dt 94 t2 dt 1 4 t 9 4t 2 98 arcsin 2 3 t 14 x 1 2 9 4x 1 2 2 98 arcsin 2 3 x 1 2 12 x 2 x 2 x 14 2 x 2 x 98 arcsin 2 3 x 1 3 C 1275 xdx x44x23 xdx x44x241 xdx x2221 x2 2 t 2xdx dt 12 dt t21 12 arctanh t 1 2 arctanhx 2 2 C 1276 cosx sin2x6sinx12 dx cosx sinx323 dx sinx 3 t cosxdx dt dt t23 1 3 arctan t 3 1 3 arctan sinx3 3 C 1277 ex 1exe2x dx ex t exdx dt dt 1tt2 dt 1 t 12 2 14 dt 3 4 t 1 2 2 arcsinh 2t1 3 arcsinh 2ex1 3 C 1278 sinxdx cos2x4cosx1 sinx cosx223 dx cosx 2 t sinxdx dt dt t23 ln t t2 3 ln cosx 2 cos2x 4cosx 1 C 1279 I lnxdx x 14 lnxln2x lnxdx x lnx2241 lnxdx x 5lnx22 lnx u dx x du I udu 5u22 u 2 z du dz z2 5z2 dz zdz 5z2 2 dz 5z2 I1 zdz 5z2 5 z2 t 2zdz dt dt 2 t t 5 z2 I1 5 u 22 1 u2 4u 1 ln2x 4 lnx I2 dz 5z2 arcsin z 5 arcsin u2 5 arcsin lnx2 5 I I1 2I2 1 ln2x 4 lnx 2arcsin lnx25 C 1280 I dx xaxb x12 4 1612 2 x1 422 3 x2 422 1 1 x3x1x24x5 ax3 b x1 cxd x24x5 |x 3x 1x2 4x 5 1 ax3 3ax2 ax 5a bx3 bx2 7bx 15b cx3 4cx2 3cx dx2 4dx 3d 1 x3a b c x23a b 4c d xa 7b 3c 4d 5a 15b 3d 1 1 1 0 0 3 1 4 1 0 1 7 3 4 0 5 15 0 3 1 3 1 5 1 1 1 0 0 0 2 7 1 0 0 8 2 4 0 0 10 5 3 1 : 4 : 5 1 1 1 0 0 0 2 7 1 0 0 2 12 1 0 0 2 1 35 1 5 1 1 1 0 0 0 2 7 1 0 0 2 12 1 0 0 0 12 1 3 5 1 5 1 1 1 0 0 0 2 7 1 0 0 2 12 1 0 0 0 12 8 5 1 5 1 1 1 0 0 0 2 7 1 0 0 2 12 1 0 0 0 12 8 5 1 5 1 1 1 0 0 0 2 7 1 0 0 0 7 12 2 0 0 0 12 8 5 1 5 1 1 1 0 0 0 2 7 1 0 0 0 152 2 0 0 0 12 8 5 1 5 15 1 1 1 0 0 0 2 7 1 0 0 0 152 2 0 0 0 152 815 5 3 1 1 1 0 0 0 2 7 1 0 0 0 152 2 0 0 0 152 24 3 1 1 1 0 0 0 2 7 1 0 0 0 152 2 0 0 0 0 26 3 d 326 ,b 1 20 ,a 1 52 ,c 2 65 I 152 dx x3 1 20 dx x1 2 65 x 3 26 x24x5 dx I 152 lnx 3 1 20 lnx 1 1 65 2x 326 65 x24x5 dx I 152 lnx 3 1 20 lnx 1 1 65 2x 152 x24x5 dx I1 165 2x4 72 x24x5 165 2x4 x24x5 72 1 65 dx x24x5 x2 4x 5 t 2x 4dx dt x 2 z dx dz I1 165 dt t 7 2 1 65 dx x221 165 ln t 7 130 dz z21 I1 165 ln t 7 130 arctan z 1 65 lnx 2 4x 5 7130 arctanx 2 I 152 lnx 3 1 20 lnx 1 1 65 lnx 2 4x 5 7130 arctanx 2 C 1294 dx x31 dx x1x2x1 1 x1x2x1 ax1 bxc x2x1 |x 1x2 x 1 1 ax2 bx2 ax bx cx c a 1 x2a b xa b c c a a b 0 a b c 0 c a 1 1 1 0 0 1 1 1 0 1 0 1 1 1 1 0 0 0 2 1 0 0 1 2 1 2 1 1 0 0 0 2 1 0 0 2 4 2 1 1 0 0 0 2 1 0 0 0 3 2 b 13 ,c 2 3 ,a 1 3 I 1 3 x1 dx 13 x 2 3 x2x1 dx 13 lnx 1 1 3 x2 x2x1 I 13 lnx 1 1 6 2x4 x2x1 I 13 lnx 1 1 6 2x1 x2x1 12 dx x2x1 I 13 lnx 1 1 6 lnx 2 x 1 12 dx x 12 2 14 1 I 13 lnx 1 1 6 lnx 2 x 1 12 dx x 12 2 34 I 13 lnx 1 1 6 lnx 2 x 1 33 arctan 2x1 3 C 1295 dx x41 dx x2122x211 dx x2122x2 dx x2 2 x1 x2 2 x1 1 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 1 1 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 d a c 0 a 2 b c 2 d 0 a b 2 c d 2 0 b d 1 1 0 1 0 0 2 1 2 1 0 1 2 1 2 0 0 1 0 1 1 : 2 1 ~ 1 0 1 0 0 1 22 1 2 2 0 1 2 1 2 0 0 1 0 1 1 ~ 1 0 1 0 0 0 22 2 2 2 0 0 2 0 2 0 0 1 0 1 1 2 2 : 2 1 ~ 1 0 1 0 0 0 1 4 2 1 0 0 1 0 1 0 0 1 0 1 1 ~ 1 0 1 0 0 0 1 4 2 1 0 0 0 4 2 2 0 0 0 4 2 0 1 ~ 1 0 1 0 0 0 1 4 2 1 0 0 0 4 2 2 0 0 0 0 2 1 d 12 ,c 2 4 ,b 1 2 ,a 2 4 I 2 4 x 1 2 x2 2 x1 2 4 x 1 2 x2 2 x1 14 2 x2 x2 2 x1 14 2 x2 x2 2 x1 I 24 x 2 x2 2 x1 24 x 2 x2 2 x1 28 2x2 2 x2 2 x1 28 2x2 2 x2 2 x1 I 28 2x 2 2 x2 2 x1 28 2x 2 2 x2 2 x1 x2 2 x 1 t 2x 2 dx dt x2 2 x 1 z 2x 2 dx dz I 28 dt t 2 8 dz z 2 8 2 x2 2 x1 28 2 x2 2 x1 I 18 2 ln t 1 8 2 ln z 2 8 dx x2 2 x1 28 dx x2 2 x1 I 18 2 ln t 1 8 2 ln z 1 4 dx x 22 2 12 1 14 dx x 22 2 12 1 I 28 ln t 2 8 ln z 2 4 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 dx x4x21 x 21 x4x21x21 dx x21 x61 dx x21 x31x31 dx x21 x31x31 ax 2bxc x31 dx 2exf x31 |x3 1x3 1 x2 1 ax2 bx cx3 1 dx2 ex fx3 1 x2 1 ax5 ax2 bx4 bx cx3 c dx5 dx2 ex4 ex fx3 f x2 1 x5a d x4b e x3c f x2a d xb e c f a d 0 b e 0 c f 0 a d 1 b e 0 c f 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 1 gauss 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 2 0 0 1 0 0 0 0 2 0 0 0 0 0 0 0 2 1 f 12 e 0 d 1 2 c 12 0 c 1 2 b 0 a 12 0 a 1 2 I 1 2 x 2 12 x31 1 2 x 2 12 x31 12 x21 x31 12 x21 x31 12 x21 x1x2x1 dx 12 x21 x1x2x1 dx I 12 x1 x2x1 dx 12 x21 x31 dx 2x11 x2x1 dx 12 x1x1 x1x2x1 dx I 2x1 x2x1 dx dx x2x1 2x11 x2x1 dx I dtt dxx 12 2 14 1 2x1 x2x1 dx dx x2x1 I ln t dx x 12 2 34 dzz dx x 12 2 14 1 I lnx2 x 1 2 33 arctan 2x1 3 3 ln z dx x 12 2 34 I lnx2 x 1 2 33 arctan 2x1 3 3 lnx 2 x 1 2 33 arctan 2x1 3 3 I ln x2x1 x2x1 2 33 arctan 2x1 3 2 33 arctan 2x1 3 C ????????? 1297 dx 1x2 2 x tan t dx 1 tan2tdt 1tan 2tdt 1tan2t2 dt 1tan2t dt 1 sin2t cos2t dt cos2tsin2t cos2t cos2tdt cos2tsin2t cos2tdt1 cos2tdt 12 1 cos2t 1 2 dt 1 2 cos2tdt 1 2 t 1 4 sin2t 1 2 arctanx 1 4 sin2arctanx 1298 I 3x5 x22x2 dx 3x5 x121 dx 3 x x121 dx 5 dx x121 3 x x121 dx I 3 x11 x121 dx 5arctanx 1 3 x1 x121 dx 3 dx x121 5arctanx 1 I 3 x1 x121 dx 3arctanx 1 5arctanx 1 3 x1 x121 dx 2arctanx 1 I1 x 1 t dx dt 3 t t21 dt t2 1 u 2tdt du 3 du2u 3 2 lnu 3 2 lnt 2 1 I1 32 ln x 1 2 1 C I 32 ln x 1 2 1 2arctanx 1 C 1299 dx x1x2x12 1 x1x2x12 ax1 bxc x2x12 dxe x2x1 |x 1x2 x 12 1 ax2 x 12 bx cx 1 dx ex 1x2 x 1 1 ax4 2ax3 3ax2 2ax a bx2 bx cx c dx4 2dx3 2dx2 dx ex3 2e x2 2ex e 1 x4a d x32a e 2d x23a b 2d 2e x2a b c d 2e a c a d 0 2a e 2d 0 3a b 2d 2e 0 2a b c d 2e 0 a c e 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 Gaussova eliminacija 1 0 0 1 0 0 0 1 0 1 2 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 e 0d 1c 0 b d 2e 0 b 1 0 b 1a 1 dx x1x2x12 dxx1 xdx x2x12 xdx x2x1 lnx 1 dx x2x1 dx x2x12 lnx 1 dx x2x1 14 1 4 dx x2x12 I lnx 1 I1 I2 I1 dx x2x1 14 1 4 dx x 12 2 34 I1 x 12 t dx dt I1 dt t2 34 2 3 arctan 2 3 t 2 3 arctan 2 3 x 12 2 3 arctan 2x1 3 I2 dx x2x12 dx x 12 2 34 2 I2 x 12 t dx dt I2 dt t2 34 2 I2 u 1 t2 34 2 dt dv 256t 4t233 dt du t v I2 t t2 34 2 256t 2 4t233 dt t t2 34 2 64 4t 233 4t233 dt I2 t t2 34 2 64 dt4t232 3 dt t2 34 3 t t2 34 2 dt t2 34 2 3 dt t2 34 3 I2 t t2 34 2 I2 3 dt t2 34 3 . . . . odusta san