１(一)含有ax+b 的积分1.C b ax a b ax d b ax a dx ++=++=+∫∫ln 1)(11b ax2.()()C u a b ax b ax d a dx u uu +++=++=+∫∫)1()()(b ax 1b ax3.C b ax b b ax a b ax b ax d a b dax bax b ax a dx b ax bb ax a dx b ax ax a dx b ax x ++−+=++−++=+−+=+=+∫∫∫∫∫)ln (1)(111222 4.++−+−++=+−−+=+=+∫∫∫∫∫∫b ax b ax d b b ax d b b ax d b ax a dx b ax b abx b ax a dx b ax x a a dx b ax x )()(2)()(12)(11232222222 C b ax b b ax b b ax a ++−+−+=ln )(2)(211223 5.()C xbax b x C a b b ax b C ax b ax b dx ax b ax b a b ax x dx ++−=++++−=+−+−= −+−=+∫∫ln 1ln ln 1ln 1ln ln 111)(11 6.()()C x b a b ax b a bx x dx b a b ax dx b a dx x b dx bx a b ax b a x b b ax x dx +−++−=−++=−++=+∫∫∫∫∫ln ln 11111222222222 C xbax b a bx +++=ln12 7.()()()()C b ax a b b ax a b ax dx a b b ax dx a dx b ax a b b ax a b ax xdx ++⋅++=+−+=+−+=+∫∫∫∫1ln 11122222 C b ax b b ax a ++++=ln 12 8.()()()()()∫∫∫∫++−+−=+−+−=+−−+=+C b ax b b ax b ax a b ax dx a b b ax xdx a b a x dx b ax a b a bx b ax a dx b ax x 2322222222222ln 212219.()()()()C x b ax b b ax b C b x b ax b b ax b x dx b b ax dx b a b ax b adxb ax x dx ++−+=+++−+=++−+−=+∫∫∫∫ln 11ln ln 1112222222 （二）含有b ax +的积分10.()()C b ax ab ax d b ax a dx b ax ++=++=+∫∫3321 11.()()()()()∫∫∫+−=++−+=+−++=+3232522315232521b ax b ax aC b ax a b b ax a dx b ax a b dx b ax b ax a dx b ax x C + 12.()()()⋅−−+=+−+−++=+∫∫∫∫b ax aa b b ax a dx b ax a b dx b ax x a b dx b ax b ax adx b ax x23152[2722127322222()()()()C b ax b abx x a aC b ax ab b ax +++−=++−+32223332381215105232]２13.()()=++−+=++−+=+−++=+∫∫∫∫∫C b ax ab b ax ab ax b ax d a b dx b ax a b ax dx a b dx b ax b ax a b ax xdx 232223211()C b ax b ax a++−=2322 14.()()()()+++−++=+−+−++=+∫∫∫∫∫∫b ax d b ax a b b ax d b ax ab ax dxab b ax xdx a b dx b ax b ax a dx b ax x 3233222222121()C b ax b bx x a ab ax dx ab +++−=+∫22232284315215.∫+bax x dx 当b>0时，有C bb ax b b ax b b ax d b b ax b b ax b ax b a b ax x dx +++−+=+++−−++=+∫∫ln 1112 当b<0时，令ax+b=t,则dx=dt a ab t d1=− C b t b b t d b t b b t d b t b t d b t t a b t dta b ax x dx +−−=−+⋅−−= +−=−=−=+∫∫∫∫∫arctan 21121122122 C b bax b+−+−=arctan 2所以=+∫b ax x dx ()()<+−+−>+++−+0arctan 10ln1b C b b ax bb C b b ax bb ax b 16.∫∫∫∫∫∫∫−+−+−=+−++=+−++=+b a dx x b bax b b ax abxb ax x dx b a dx bx b ax b ax b ax x dx b a dx bax bx b ax b ax xdx 2222222222222∫∫∫+−+−=+−′−′−=+b ax x dx b a bx b ax b ax x dx b a dx v v u v u bax x dx 22217.∫∫∫∫∫+++=+++=+++=+b ax x dx b b ax b ax x dx b dx bax a dx b ax x b b ax a dx x b ax 2)(18.∫∫∫∫∫+⋅−+−=+++=+++=+bax x dx b a b bx b ax b b ax x dx a b ax x dxb dx b ax x b b ax x a dx x b ax 21)(222３∫∫+++−=++bax x dxa xb ax b ax x dx a2（三）含有x 2±a 2的积分 19.∫+22a x dx设x=atant(22ππ<<−t )那么t a a x2222sec =+ tdt a dx 2sec = 于是C ax a C dt a dt t a t a a x dx +=+==+∫∫∫arctan 11sec sec 22222 20.()∫+na x dx22用分部积分法，n>1时有()∫+na xdx22()()()()()()∫∫+−+−++=+−++=−−−dx a x a a x n a x xdx a x x n a x xnn n n n 222122122222122112)1(2 即 ()()()n n n n I a I n a xxI 21122112−−++=−−− 于是()()()−++−=−−1122232121n n nI n ax xn a I由此作递推公式并由 C a xa I +=arctan 11即可得n I ()()()()()∫∫−−+−−++−=+∴12221222221232121n n n axdxa n n a x a n a x dx21.C ax ax a dx a x a x a dx a x a x a x dx =+−= +−−=−⋅+=−∫∫∫ln 2111211122 (四)含有ax 2+b(a>0)的积分22.()∫∫∫>+=+=+=+0arctan 11111222b C x b a ab x b a x badb a b x b a dx b b ax dx()()()0ln 2111212<+−+−−−=−+−−−−=−−−+=+∫∫∫b C b x a b x a abdx b x a b x a ab b x a b x a dx b ax dx23.∫∫++=+=+C b ax b ax dx b ax xdx 2222ln 2121 24.()()C b ax x b C b ax a b a x b dx bax b axbx dx x b ax ++=++⋅−=+−=+∫∫22222ln 21ln 21ln 111 25.∫∫∫∫+−=+−++=+b ax dx a b a x dx b ax a b dx bax b ax a b ax dx x 22222211 26.()∫∫∫∫++−−=+−=+C b ax dx b abx b ax dx b a bx dx b ax x dx 222221４27.()()ln 2ln 221ln 21ln 2112222222222222323b a x b a bx C b ax a b a x b a bx dx x b a b ax b x a bx b ax x dx +−−=++⋅+−−=−++=+∫∫ C bxx b ax b a C b ax +−+=++2222221ln 2 28.()()()()()()∫∫∫∫∫∫+++−+′=+++−=+++−=+b ax dx b dx b ax axu b ax u b b ax dx b dx b ax ax b b dx bax b b ax b ax b b ax dx22222222222222212212121212222ax b axu u b x u a −=−′+′ b b u =′ 1=′u x u =()()()∫∫∫∫+++⋅=+++⋅−+′=+bax dxb b ax x b b ax dx b dx baxxax b ax x baxdx222222222121212 (五)含有)0(2>++a c bx ax的积分29.dx a ac b b a x a c bx ax dx 1222442−∫∫−− +=++ 当ac b 42<时，有 dx a ac b b a x a c bx ax dx 1222442−∫∫−− +=++∫+−+−=142244222b ac a b x a dx b ac a 令2422b ac a b x a t − += 则 dx b ac a dt 242−=则原式=C b ac a b x a b ac C t b ac t dt a b ac b ac a +−+−=+−=+−⋅−∫222222422arctan 42arctan 4112444当ac b 42>时有 dx ac b a b x a b ac a c bx ax dx ∫∫−+−−−=++42414422222 令ac b a b x a t 4222− += 则dx ac b a dt 422−= 则原式C acb b ax ac b b ax ac b t dta acb ac b a +−++−−+−=−−−=∫4242ln 4112444222222 综上所述()>+−++−−+−<+−+−=++∫)4(4242ln 41442arctan 4222222222ac b C ac b b ax ac b b ax ac b ac b C b ac b ax b ac c bx ax dx５30.()()c bx ax ac bx ax dx a b dx c bx ax b ax a c bx ax a bdx dx c bx ax a b ax c bx ax xdx ++=++−+++=++−+++=++∫∫∫∫∫222222ln 212221222∫++−c bx ax dx a b 22 其中 ()[]()()+++=++′++=′++c bx ax b ax c bx ax c bx ax c bx ax 22222ln （六）含有()022>+a a x 的积分31.∫+22a x dx 由于t t 22sec tan1=+，不妨设 <<−=22tan ππt t a x ，那么t a a x sec 22=+，tdt a dx 2sec =于是∫+22ax dx∫∫==tdt dt t a ta sec sec sec 2，利用例17的结果得C t t a x dx ++=+∫tan sec ln 22 作图可知axt =tan ，aa x t 22sec +=，且0tan sec >+tt ，因此()C a x x C a a x a x a x dx+++=+++=+∫2212222ln ln 32.()∫+322axdx设<<−=22tan ππt t a x ，那么ta a x sec 22=+，tdta dx 2sec =，于是()C t adt t a dt t a t a axdx+−===+∫∫∫sin 1sec 11sec sec 22332322axt =tan ，txa a t cos sec 122=+=，22cos tan sin ax x t t t +=⋅= ，()C ax ax axdx++−=+∫22232233.∫+22a x xdx ，不妨设 <<−=22tan ππt t a x，那么t a a x sec 22=+，tdt a dx 2sec =，于是∫+22a x xdx =∫∫++=+==C a x C t a tdt t a tdt a t a t a 222sec tan sec sec sec tan34.()∫+322a x xdx设 <<−=22tan ππt t a x，那么()t a a x33322sec =+，tdt a dx 2sec =，于是()C ax Ct a dt t t a tdt a t a t a a xxdx++−=+−===+∫∫∫222333221cos 1sec tan 1sec sec tan 35.−+=+++−++++=+−+=+∫∫∫2222222222222222222)ln()ln(22a x x C a x x a a x x a a x x a x dxa dx a x ax dx x ()C a x x a +++222ln 2６36.()()()()C ax x a x x axdxa ax dx dx axa a x axdxx ++−++=+−+=+−+=+∫∫∫∫22223222223222223222ln37.∫+22a x xdx 设 <<−=22tan ππt t a x，那么()t a a x33322sec =+，tdt a dx 2sec =，于是C x aa x a C t t a t dt a t a t a tdt a ax x dx+−+=+−==⋅=+∫∫∫22222ln 1cot csc ln 1sin 1tan sec sec38.∫+222a x xdx设 <<−=22tan ππt t a x，那么()t a a x33322sec =+，tdt a dx 2sec =，于是C x a x a C t a dt t t a t a t a tdt a ax xdx++−=+−==⋅=+∫∫∫22222222222sin 1sin cos 1sec tan sec 39.∫+dx a x 22 设 <<−=22tan ππt t a x ，那么()t a a x33322sec =+，tdt a dx 2sec =，于是−⋅=⋅−⋅===⋅=+∫∫∫∫∫t t a tdt t a t t a t td a tdt a tdt a t a dx a x tan sec tan sec tan sec tan sec sec sec sec 2222232222()∫∫+++−⋅=−1232222tan sec ln sec tan sec 1secsec C t t a tdt a t t a dt t t a()C a x x a a x x C t t t t a dx a x +++++=++++=+∴∫22222222ln 22tan sec ln tan sec 240.()∫+dx a x322设 <<−=22tan ππt t a x ，那么()t a a x33322sec =+，tdt a dx 2sec =，于是()∫∫=+tdt a dx a x54322sec ∫∫∫⋅−==tdt t t t t t t td tdt tan sec sec tan 3tan sec tan sec sec 2335∫∫∫+−=−=tdt tdt t t tdt t t t 353233sec 3sec 3tan sec tan sec 3tan sec()13tan sec ln tan sec 21sec C t t t t tdt ++++=∫()1535tan sec ln tan sec 23sec 3tan sec sec C t t t t tdt t t tdt +++++−=∴∫∫()()()ax a a xa C t t t t a t t a tdt a dx a x ⋅+=++++==+∴∫∫33224143454224tan sec ln tan sec 83tan sec 4sec ()()C a x x a a x a x x C a a x x a x a x a ++++++=++++++22422221222224ln 83528ln 8341.dx a x x ∫+22 设 <<−=22tan ππt t a x ，那么()t a a x33322sec =+，tdt a dx 2sec =，于是７()C a xC t a t t d a tdt t a tdt a t a t a dx a x x ++=+=−==⋅⋅=+∫∫∫∫32233433322231cos 3cos cos sec tan sec sec tan42.()[]()()22222223222222222222528a x a x x dx a x a dx a xdx a x a a x a x dx a x x ++=+−+=+−++=+∫∫∫∫()()()()C a x x a a x a x xC a x x a a x x a a x x a +++−++=+++++−+++2242222222222224ln 828ln 22ln 8343.dx x a x ∫+22 设 <<−=22tan ππt t a x ，那么()t a a x33322sec =+，tdt a dx 2sec =，于是ln cot csc ln cos sin cos sin cos sin sec tan sec 2222222a x a C t t a tat dt a dt t t a t t dt a tdt a t a t a dx x a x ++=+−+=+===+∫∫∫∫∫C xaa x +−+2244.dx x a x ∫+222 设 <<−=22tan ππt t a x ，那么()t a a x33322sec =+，tdt a dx 2sec =，于是C tt t t t d tdt t t dt tdt t tdt a t a t a dx x a x +−+=+====+∫∫∫∫∫∫sin 1tan sec ln sin sin sec cos sin cot sec sec tan sec 2223222222 ()C xa x a x x ++−++=2222ln（七）含有()022>−a a x 的积分45.∫−22a x dx 当a x>时，设<<=20sec πt t a x ，那么t a t a a x tan 1sec 222=−=−，tdt t a dx tan sec =，于是()()C a x x C a ax a x C t t tdt dt t a t t a a x dx+−+=+−+=++===−∫∫∫2212222ln ln tan sec ln sec tan tan sec 当a x−<时，令u x −=，那么a u >，由上段结果有()()221222222ln ln a x x C a u u au du ax dx −+−−==+−+−=−−=−∫∫()C a x x C a a x x C xa x C +−−−=+−−−=+−−=+2212221221ln ln 1ln 综上所述，C a x x a x dx +−+=−∫2222ln８46.()∫−322a xdx，设<<=20sec πt t a x，则()t a a x33322tan =−，tdt t a dx tan sec =，于是()C ax a x C t a dt t t a dt t t a dt t a t t a a xdx+−−=+−====−∫∫∫∫2222222233322sin 1sin cos 1tan sec 1tan tan sec47.∫−22a x xdx ，设<<=20sec πt t a x，则t a a x tan 22=−，tdt t a dx tan sec =，于是∫∫∫+−=+===−C a x C t a tdt a tdt t a ta ta a x xdx 22222tan sec tan sec tan sec48.()∫−322a x xdx,设<<=20sec πt t a x，则t a a x tan 22=−，tdt t a dx tan sec =，于是()∫∫∫+−−=+−===−C ax C t a dt t a tdt t a t a t a a xxdx222333221cot 1sin 11tan sec tan sec 49.C a x x a a x x a a x x a x dxa dx a x a x a x dx x +−++−+−−=−+−−=−∫∫∫2222222222222222222ln ln 212C a x x a a x x +−++−=222222ln 2250.()()()()−−+−+=−+−=−+−−=−∫∫∫∫∫22222232222232223222232221ln a x x a a a x x a xdxa a x dx a xdxa dx a xa x a xdxx C ax xa x x C+−−−+=+2222ln51.∫−22a x x dx ,设<<=20sec πt t a x，则t a a x tan 22=−，tdt t a dx tan sec =，于是，当0>x 时有C x a C a t dt tt a tt a a x xdx +=+==−∫∫arccos tan sec tan sec 22当0<x时有，C x aa a x x dx +−=−∫arccos 122，综上所述，有C x a aa x x dx +=−∫arccos 122 52.∫−222a x xdx，设<<=20sec πt t a x，则t a a x tan 22=−，tdt t a dx tan sec =，于是C xa a x C t a t dt a dt t a t a t t a a x xdx+−=+==⋅=−∫∫∫2222222222sin 1sec 1tan sec tan sec９53.dx a x ∫−22,设<<=20sec πt t a x ，则t a a x tan 22=−，tdt t a dx tan sec =，于是−++=−==⋅=−∫∫∫∫t t a t t a dt t t a tdt t a tdt t a t a dx a x tan sec ln 2tan sec 2cos cos 1tan sec tan sec tan 223222222()C a x x a a x x C t t a +−+−−=++222222ln 22tan sec ln54.()dx a x∫−322,设<<=20sec πt t a x ，则t a a x tan 22=−，tdt t a dx tan sec =，于是()∫∫∫==−tdt t a tdt t ta a dx a xsec tan tan sec tan 4433322()∫∫∫∫∫+−=−=tdt tdt tdt tdt t dt t t sec sec 2sec sec 1sec sec tan 35224()∫+++++=135tan sec ln tan sec 83tan sec 41sec C t t t t t t tdt ()23tan sec ln tan sec 21sec C t t t t tdt +++=∫;∫++=3tan sec ln sec C t t tdt()+⋅−++++=+−=∴∫∫∫∫t t t t t t t t tdt tdt tdt tdt t tan (sec 212tan sec ln tan sec 83tan sec 41sec sec 2sec sec tan 3354a a x a x a a x a x a a x ax C C C t t t t 22222233321ln 8385412tan sec ln )tan sec ln −++−⋅⋅−−⋅⋅=+−++++3212C C C +−+ ()()Ca x x a a x a x x C a x x a a x x a a x x dx ax+−++−−=+−++−−−=−∫2242222224222223322ln 83528ln 8385455.dx a x x ∫−22,设<<=20sec πt t a x ，则t a a x tan 22=−，tdt t a dx tan sec =，于是 ∫∫∫∫∫∫−=−==⋅⋅=−t dt a t dt a dt t t a tdt t a tdt t a t a t a dx a x x 234342322322cos cos cos cos 1sec tan tan sec tan sec()()C a xC t a t a t d t a t a t td a +−=+=−+=−=∫∫3223332332331tan 3tan tan tan 1tan tan sec56.()()222224222222222222222ln 83528a x x a a x x a a x a x xdx a x adx a x ax dx a x x−⋅+−++−−=−+−−=−∫∫∫ ()C a x x a a x a x xC a x x a +−+−−−=+−+−2242222224ln 828ln 257.dx x a x ∫−22, 设<<=20sec πt t a x ，则t a a x tan 22=−，tdt t a dx tan sec =，于是当0>x 时，有１０()C xaa x C t t a dt t a tdt a tdt t a t a t a dx x a x +−−=+−=−===−∫∫∫∫arccos tan 1sec tan tan sec sec tan 222222 当0<x时，有C xa a x dx x a x +−−−−=−∫arccos )(2222；综上所述，C x a a x dx x a x +−−=−∫arccos 2222 58.dx x a x ∫−222，设 <<=20sec πt t a x ，则t a a x tan 22=−，tdt t a dx tan sec =，有∫∫∫∫===−t td dt t t tdt t a ta t a dx x a x sin tan sec tan tan sec sec tan 2222222，令t u sin =，则 C ttt C u u u u du u du u du u du du u u t td +−++−=+−++−=++−+−=−+−=−=∫∫∫∫∫∫sin 1sin 1ln 21sin 11ln 21121121111sin tan 2222()C a x x xa x +−++−−=2222ln（八）含有()022>−a a x 的积分59.C a xa x dxa x a dx += −=−∫∫arcsin 11222 60.()∫−322x a dx,令 <<−=22sin ππt t a x，则()t a x a33322cos =−，tdt a dx cos =，于是()∫∫∫+−=+===−C xa a x C t a t dt dt t a t a x adx2222233322tan 1cos cos cos 61.∫−22x a dx ，令 <<−=22sin ππt t a x，则t a x a cos 22=−，tdt a dx cos =，于是∫∫∫+−−=−=−==−C x a t a tdt a tdt a ta ta x a dx 2222cos sin cos cos sin62.()∫−322x axdx,令 <<−=22sin ππt t a x，则()t a x a33322cos =−，tdt a dx cos =，于是()∫∫∫+−=+===−C xa C t a dt t t a tdt a t a t a x axdx222333221cos 1cos sin 1cos cos sin 63.∫−222x a dx x ,令 <<−=22sin ππt t a x，则t a x a cos 22=−，tdt a dx cos =，于是C x a x a x a C t a t a tdt a tdt a t a t a x a dxx +−−=+−===−∫∫∫2222222222222arcsin 22sin 421sin cos cos sin64.()∫−3222x a dxx ,令 <<−=22sin ππt t a x，则()t a x a33322cos =−，tdt a dx cos =，于是()C a xxa x C t t dt t dt tdt a t a t a x adxx +−−=+−=−==−∫∫∫∫arcsin tan cos cos cos sin 22233223222 65.∫−22x a xdx ,令 <<−=22sin ππt t a x，则t a x a cos 22=−，tdt a dx cos =，于是C x x a a a C x x a a x a C t t a dt t t a t a xa x dx+−−=+−−=+−==−∫∫2222222ln 1ln 1cot csc ln 1cos sin cos 66.∫−222x a x dx,令 <<−=22sin ππt t a x，则t a x a cos 22=−，tdt a dx cos =，于是C x a x a C t a t dt a t ta a tdt a xa x dx+−−=+−===−∫∫∫22222222222cot 1sin 1cos sin cos 67.∫−dx x a 22,令 <<−=22sin ππt t a x ，则t a x a cos 22=−，tdt a dx cos =，于是C x a x a x a C t a t a dt t a tdt a tdt a t a dx x a +−+=++=+==⋅=−∫∫∫∫22222222222arcsin 22sin 4222cos 1cos cos cos68.()dx x a∫−322,令 <<−=22sin ππt t a x ，则()t a x a33322cos =−，tdt a dx cos =，于是()()=++=+==⋅=−∫∫∫∫∫∫dt t a tdt a t a dt t atdt atdt a t adx xa2cos 42cos 2442cos 1coscos cos 2444244433322()=+−−+−+=++++C x a x a x x a x a a x a C t a t a t a a x a 822arcsin 834sin 3282sin 4arcsin 4222222244444()C a x a x a x a x ++−−arcsin 832584222269.dx x a x ∫−22,令 <<−=22sin ππt t a x ，则t a x a cos 22=−，tdt a dx cos =，于是 ()C x aC t a t td a tdt a t a t a dx x a x +−−=+−=−=⋅⋅=−∫∫∫32233232231cos 3cos cos cos cos sin70.dx x a x ∫−222,令 <<−=22sin ππt t a x ，则t a x a cos 22=−，tdt a dx cos =，于是()∫∫∫∫∫∫−=−==⋅⋅=−tdt a tdt a dt t t a tdt t a tdt a t a t a dx x a x442422422422222sin sin sin 1sin cos sin cos cos sin()∫∫∫∫+−=−+−−=−−−=dt t a t a tdt a t a t a t a t a dt t a dt t a 24cos 1442cos 42sin 4412sin 42142cos 122cos 144244444244()C x a x a xa x a C t a t a +−−−=+−=222244428arcsin 84sin 32871.dx x x a ∫−22,令 <<−=22sin ππt t a x ，则t a x a cos 22=−，tdt a dx cos =，于是∫∫∫∫∫∫=++−=−=−===−C t a t t a tdt a t dta dt t t a tdt t a tdt a ta t a dx x x a cos cot csc ln sin sin sin sin 1cos cot cos sin cos 222C x a xx a a a C a x a a x x a x a a +−+−−=+−+−−22222222ln ln72.dx x x a ∫−222,令 <<−=22sin ππt t a x ，则t a x a cos 22=−，tdt a dx cos =，于是C a x x x a C t dt tdt dt t t tdt tdt a t a t a dx x x a +−−−=+−−=−=−==−∫∫∫∫∫∫arcsin cot sin sin sin 1cot cos sin cos 22222222222 （九）含有()02>++±a c bx ax 的积分73.∫∫−++=++2222421a b a c a b x dxacbx ax dx，令t abx =+2，则dt dx = 当042>−ac b 时，则令()044222>=−u u a ac b ，则∫∫∫−=−++=++2222221421ut dt aa b a c a b x dxa c bx ax dx再令r u tsec =,rdr r u dt sec tan =,r u u t tan 22=−,于是∫∫∫++===−122tan sec ln 1sec 1tan sec tan 11C r r ardr a dr r u r r u a u t dt aac b b ax a ac b a bx utr 42442sec 222−+=−+== ; acb cbx ax a u u t r r r 412cos cos 1tan 22222−++=−=−=C c bx ax a b ax a C ac b c bx ax a b ax a c bx ax dx+++++=+−++++=++∫2122222ln 1422ln1当042<−ac b时，则令u a b ac =−242，a b x t 2+=，∫∫∫+=−++=++2222221421ut dtaa b a c a b x dxa cbx ax dx令r u t tan =，rdr u dt 2sec =∫∫∫++===+1222tan sec ln 1sec 1sec sec 11C r r a rdr a r u rdr u a u t dtaa c x ab x a b ac a b ac a b x a b ac u t u r ++−=−++−=+=22222222212444244sec 1；2242242tan b ac b ax a b ac a bx u t r −+=⋅−+==C c bx ax a b ax a C b ac b ax b ac c bx ax a a c bx ax dx+++++=+−++−++=++∴∫21222222ln 14242ln 1综上所述，C c bx ax a b ax acbx ax dx +++++=++∫2222ln 174.dx a b ac a b x a dx c bx ax 2222442−++=++∫∫ 当042>−ac b 时，令a b x t 2+=，u a ac b =−242 ∫∫−=++dt u t a dx c bx ax 222 ，再令r u t sec =，rdr r u dt tan sec =()C r r u a r r r r u a rdr r u a dt u t a +⋅−++⋅=⋅=−∫∫tan sec ln tan sec ln tan sec 21sec tan 222222−−⋅++⋅−+−=++⋅−⋅=acb c bx ax a acb b ax a ac b aC r r u a r r u a 412422421tan sec ln 21tan sec 2122222122C c bx ax a b ax a b ac c bx ax a b ax C acb c bx ax a b ax a ac b a +++++−++++=+−++++−23221222222ln 8442422ln 442 当042<−ac b 时，令abx t 2+=，u a b ac =−242 ；∫∫+=++dt u t a dx c bx ax 222；再令r u t tan =，rdru dt2sec =，于是()×−⋅++−⋅=++⋅==++∫∫2222232242442tan sec ln tan sec 21sec b ac a a c x a b x a b ac a r r r r u a rdr u a dx c bx axln 84424242ln 4421423221222222ab ac c bx ax a b ax C b ac b ax b ac c bx ax a a b ac a b ac bax −++++=+−++−++−⋅+−+C c bx ax a b ax +++++222综上所述，∫+++++−++++=++C c bx ax a b ax a b ac c bx ax a b ax dx c bx ax 2322222ln 844275.∫∫−++=++2222442a b ac a b x a xdxcbx ax xdx ；令a bx t 2+=，当0>∆时，令2244a b ac u −=122222222222ln 22121C u t t a a ba u t u t a dt ab dt u t ta dt u t a bt a cbx ax xdx+−+−−=−−−=−−=++∫∫∫∫C c bx ax a b ax aa b a c bx ax C a c x a b x a b x a a b a c x a b x a+++++−++=+++++−++=2212222ln 22ln 21当0<∆时，令abac u 242−=，于是∫∫∫−+=+−=++aa b u t a tdt dt u t a a bt cbx ax xdx 222222212212222222ln 2ln 21C a c x a b x a b x aa b a a cx a b x C u t t aa b u t a u t dt +++++−++=+++−+=+∫C c bx ax b ax aa a c bx ax +++++−++=222ln 21综上所述，C c bx ax b ax a a ba c bx ax c bx ax xdx+++++−++=++∫2222ln 2 76.∫∫−−−=−+22222441a b x a ac b dx a ax bx c dx；令a bx t 2−=，2244a b ac u +=，于是C acb b ax a C t ua t u dt aax bx c dx ++−=+−=−=−+∫∫42arcsin 1arcsin 11222277.∫∫−−+=−+2222244a b x a ac b a dx ax bx c ；令t abx =−2；2244a ac b u +=；于是+⋅−+−=++−=−=−+∫∫ax x a ba c ab x C u tau t u a tdt t u a dx ax bx c 1221arcsin 222222222C acb b ax a a ac b ax bx c a b ax C ac b b ax a a ac b ++−++−+−=++−+42arcsin 844242arcsin 8422222 78.∫∫−−+=−+22222441a b x a ac b xdxa ax bx c xdx；令a bx t 2−=；2244a b ac u +=；于是C acb b ax a a bt u a dt t u aa bdt t u t a dt t u a bt aax bx c xdx ++−+−−=−+−=−+=−+∫∫∫∫42arcsin 21121212222222222C ac b b ax a a b x bx c a C ac b b ax a a b x x a b a c a++−+−+−=++−+−+−=42arcsin 2142arcsin212222 （十）含有bx ax −−±或()()x b a x −−的积分79.∫−−dx bx a x ；令b x ax t −−=；则1−−=t abt x；()dt t ba dx 21−−=；于是()()()()()()∫∫∫∫∫−−+−−=−−=−−=−−=−−t d t b a t t a b t d t a b dt t t b a dt t b a t dx b x a x 111111122 ()()()()()11121121111121−−=−−−−++−+−−=+−−−+−−=∫∫∫t t a b t t d a b t t d a b t t a b t d t t b a t t a b()()()()()a x a b bx ax b x C b x a x a b a b b x a x b x a x a b C t t a b −−+−−−=+ −+−−−+−−⋅−−−=+−+−+ln(ln 211ln 2121()()()C b x a x a b bx ax b x C b x +−+−−+−−−=+−+ln )180.dx x b a x ∫−−；令t x b a x =−−；则t a bt x ++=1；()dt t ab dx 21+−=；于是()()()()()()()C t b a t t b a t t d b a t t b a t d t b a dt t t a b dx x b a x +−−+−=+−−+−=+−=+−=−−∫∫∫∫arcsin 1111112()()C xb ax a b x b a x x b +−−−+−−−=arctan81.()()∫−−x b a x dx;令t a x =−；则t a x +=；dt dx =；于是()()()()∫∫∫−+−=−−=−−ta b t dtt a b t dt x b a x dx 2()∫−+−−=2224b a t ba dt ；令02>=−u a b ；2ba t r −+=则()C ab ba x r u drb a t b a dt +−−−=−=−+−−∫∫2arcsin24222282.()()∫−−dx x b a x ；令t a x =−；则t a x +=；dt dx =；于是()()()∫∫−+−−=−−dt b a t b a dx x b a x 2224令02>=−u a b ；2ba t r −+=；则()()42arcsin 2222222b a x C ur u r u r dr r u dx x b a x −−=++−=−=−−∫∫ ()()()C ab b a x a b dx x b a x +−−−−+−−2arcsin 82(十一)含有三角函数的积分 83.∫+−=C x xdx cos sin 84.∫+=C x xdx sin cos 85.∫∫+−==C x dx xxxdx cos ln cos sin tan 86.∫∫+==C x dx xxxdx sin ln sin cos cot 87.C x x x d x x x d x x x d x dx xdx ++=+ += + ++= + ++==∫∫∫∫∫42tan ln 42tan 42tan 42cos 42tan 422cos 2sin 22cos sec 2πππππππππ x x x x x x x x x cot csc sin cos 1sin 2sin 22cos 2sin2tan 2−=−=== ; =++− +=∫C x x xdx 2cot 2csc ln sec ππ C xx ++tan sec ln88.C x x C x x xd x x dx x x dx x dxxdx +−=+=====∫∫∫∫∫cot csc ln 2tan ln 2tan 2tan2cos2tan 2cos 2sin 2sin csc 2 89.C x xdx +=∫tan sec 290.C x xdx +−=∫cot csc 2 91.∫+=C x xdx x sec tan sec 92.∫+−=C x xdx x csc cot csc93.C x x dx x xdx +−=−=∫∫2sin 41222cos 1sin 2 94.C x x dx xxdx ++=+=∫∫2sin 41222cos 1cos 295.()∫∫∫∫−−−−−−+−=+−=−=xdx x n x x xdx xd x x x xd xdx n n n n n n 221111cos sin 1sin cos sin cos sin cos cos sin sin ()()()()∫∫∫−−−+−=−−+−=−−−−xdx n xdx n x x xdx x n x x n n n n n sin 1sin 1sin cos sin sin 11sin cos 21221∫∫−−−+−=∴xdx n n x x n xdx n n n 21sin 1sin cos 1sin 96.()+=−+=−==−−−−−−∫∫∫∫x x xdx x n x x x xd x x x xd xdx n n n n n n n 1221111cos sin cos sin 1cos sin cos sin cos sin sin cos cos()()()()∫∫∫−−−+=−−−−−xdx n xdx n x x xdx x n n n n n cos 1cos 1cos sin cos cos 112122∫∫−−−+=∴xdx n n x x nxdx n n n 21cos 1cos sin 1cos 97.∫∫∫∫−−−−−−−++===xdx n n x x n xdx n xdx xdx n n n nn 211sin 1cos sin 1sin 1sin sin ∫∫−−+−=−−x dx n n x n x dx n n n sin 21sin cos 21sin 12 ∫∫−−−−+−−=∴x dx n n x x n xdx n n n 21sin 12sin cos 11sin 98.()∫∫∫∫∫=+−=−===−−−−−−−−−−−xdx x n x x x xd x x x xd xdx xdx n n n n n n n 221111sin cos 1sin cos cos sin sin cos sin cos cos cos ()()∫∫−−−−+++−xdx n xdx n x x n n n cos 1cos 1sin cos21∫∫−−−−++−=∴=xdx n n x x n xdx n n n 21cos 1sin cos 1cos 将n 换成2−n 有∫∫−−+−−=−−x dx n n x x n x dx n n n cos 21cos sin 21cos 12 ∫∫−−−−+−−=∴x dx n n x xn xdx n n n 21cos 21cos sin 11cos 99.∫∫∫∫−+−++−−+−+=+==x xd n x x n x xd n x xd x xdx x m n m n n m n m nm 1111111cos sin 11cos sin 11sincos 11sin sin cos sin cos ()∫∫−+−++=+−++=−+−−++−dx x x x n m x x n xdx x n m x x n m n n m m n n m 22112211cos 1cos sin 11sin cos 11cos sin 11sin cos 11 ∫∫+−−+−++=−+−xdx x n m xdx x m m x x n nn m n n m cos sin 11cos sin 11sin cos 11211 ∫∫−+−+−++=+−+∴xdx x n m x x n xdx x n m n m n m nm sin cos 11sin cos 11sin cos 111211 ∫∫−+−+−++=∴xdx x nm m x x m xdx x n m n m nm sin cos 1sin cos 11sin cos 211 又有∫∫∫∫++−+−+++−=+−=−=x m x x m x xd m x xd x xdx x m m n m n n m n m 11111cos 11cos sin 11cos sin 11cos sin cos sin cos()∫∫−+−++−=+−++−=−+−−++−−x x x m n x x m xdx x m n x x m x d n m m n n m m n n 221122111sin 1sin cos 11cos sin 11sin cos 11cos sin 11sin∫∫+−−+−++−=−+−xdx x m n xdx x m n x x m dx n m n m m n sin cos 11sin cos 11cos sin 11211。