1、(一)含有 ax+b 的積分 1.cbax a baxd baxa dx +=+ + = + ln 1 )( 11 bax 2.()()c ua bax baxd a dx u uu + + + =+=+ ) 1( )( )(bax 1 bax 3.cbaxbbax abax baxd a b dax bax bax a dx bax bbax a dx bax ax a dx bax x += + + + + = + + = + = + )ln( 1)(111 222 4. + + += + + = + = + bax baxd bbaxdbbaxdbax a dx bax babxbax

2、a dx bax xa a dx bax x)( )(2)()( 12)(11 2 3 22 2 22 2 2 cbaxbbaxbbax a + +=ln)(2)( 2 11 22 3 5.()c x bax b xca b bax b caxbax b dx axbaxb a baxx dx + + =+=+= + = + ln 1 lnln 1 ln 1 lnln 111 )( 11 6. ()() cx b a bax b a bxx dx b a bax dx b a dx xb dx bx a baxb a xbbaxx dx += + += + += + lnln 11111 22

3、22 2 2 2 22 c x bax b a bx + + +=ln 1 2 7. ()()()() c baxa b bax abax dx a b bax dx a dx baxa b baxabax xdx + + += + + = + + = + 1 ln 111 22222 c bax b bax a + + +=ln 1 2 8. () () ()()() + + += + + = + + = + c bax b baxbax abax dx a b bax xdx a b a x dx bax a b a bx bax a dx bax x 2 322 2 222 2 2 2

4、2 2 ln2 12 21 9. ()()()() c x bax bbaxb c b x bax bbaxbx dx bbax dx b a baxb adx baxx dx + + + =+ + =+ + + = + ln 11ln ln 111 2222222 （二）含有bax+的積分 10.()()cbax a baxdbax a dxbax+=+=+ 3 3 21 11.()()()() () +=+=+=+ 3 2 3 2 5 2 23 15 2 3 2 5 21 baxbax a cbax a b bax a dxbax a b dxbaxbax a dxbaxxc+ 12.()

5、()()+=+=+ bax aa b bax a dxbax a b dxbaxx a b dxbaxbax a dxbaxx23 15 2 2 7 221 2 7 32 2 2 2 2 ()()()()cbaxbabxxa a cbax a b bax+=+ 3 222 3 3 3 2 3 81215 105 2 3 2 13. () ()=+= + + += + + + = + cbax a b bax abax baxd a b dxbax abax dx a b dx bax bax abax xdx 2 3 22 2 3 211 ()cbaxbax a +=2 3 2 2 14. (

6、) ()()()+= + + + + = + baxdbax a b baxdbax abax dx a b bax xdx a b dx bax bax a dx bax x 3 2 3 32 2 2 2 2 2121 ()cbaxbbxxa abax dx a b += + 222 32 2 843 15 2 15. +baxx dx 當(dāng) b0 時(shí)，有 c bbax bbax b baxd bbaxbbaxbaxb a baxx dx + + + =+ + + = + ln 111 2 當(dāng) b0 時(shí)，令 ax+b=t,則 dx=dt aa bt d 1 = c b t bb t d b t

7、 b b td b t btd bt t a bt dt a baxx dx + = + = + = = = + arctan 2 1 12 1 1 2 2 1 22 c b bax b + + =arctan 2 所以= + baxx dx () () + + + 0arctan 1 0ln 1 bc b bax b bc bbax bbax b 16. + + = + + + = + + + = +b a dx xb baxb bax abx baxx dx b a dx bx bax bax baxx dx b a dx baxbx bax baxx dx 2 2 2 2 2 22 2

8、22222 + + = + = +baxx dx b a bx bax baxx dx b a dx v vuvu baxx dx 22 2 17. + += + + + = + + + = + baxx dx bbax baxx dx bdx bax a dx baxx b bax a dx x bax 2)( 18. + + = + + + = + + + = + baxx dx b ab bx bax b baxx dx a baxx dx bdx baxx b baxx a dx x bax 2 1 )( 2 2 2 + + + = + + baxx dxa x bax baxx dx

9、 a 2 （三）含有 x 2a2的積分 19. + 22 ax dx 設(shè) x=atant( 22 1 時(shí)有 () + n ax dx 22 ()()() () ()() + + + + = + + + = dx ax a ax n ax x dx ax x n ax x nnnnn 22 2 1 22 1 2222 2 1 22 1 12) 1(2 即 () ()() nnnn iain ax x i 2 11 22 1 12+ + = 于是 () () () + + = 11 22 2 32 12 1 nnn in ax x na i 由此作遞推公式并由 c a x a i+=arctan

10、1 1 即可得 n i ()()() () () + + + = + 1 22 21 22222 12 32 12 1 nnn ax dx an n axanax dx 21.c ax ax a dx axaxa dx axaxax dx = + = + = + = ln 2 111 2 111 22 (四)含有 ax 2+b(a0)的積分 22.() += + = + = + 0arctan 1 1 1 1 1 2 2 2 bcx b a ab x b a x b a d b a b x b a dx bbax dx ()() ()0ln 2 111 2 1 2 +acbxax的積分 29.

11、dx a acb b a xa cbxax dx 1 2 2 2 4 4 2 += + 當(dāng)acb4 2 時(shí)有 dx acb a b xa bac a cbxax dx + = + 4 2 4 1 4 4 2 2 2 2 2 令 acb a b xa t 4 2 2 2 + = 則dx acb a dt 4 2 2 = 則原式c acbbax acbbax acb t dt a acb acb a + + + = = 42 42 ln 4 1 12 4 4 4 2 2 2 2 2 2 綜上所述 () + + + +aax的積分 31. + 22 ax dx 由于tt 22 sectan1=+，不

12、妨設(shè) +tt， 因此()caxxc a ax a x ax dx +=+ + += + 22 1 22 22 lnln 32. () + 3 22 ax dx 設(shè) = 22 tan ttax，那么taaxsec 22 =+，tdtadx 2 sec=，于是 () ct a dt ta dt ta ta ax dx += + sin 1 sec 11 sec sec 2233 2 3 22 a x t =tan，t xa a t cos sec 1 22 = + =， 22 costansin ax x ttt + = ， () c axa x ax dx + + = + 2223 22 33.

13、 + 22 ax xdx ，不妨設(shè) = 22 tan ttax，那么taaxsec 22 =+，tdtadx 2 sec=，于是 + 22 ax xdx = +=+=caxctatdttatdta ta ta 222 sectansecsec sec tan 34. () + 3 22 ax xdx 設(shè) = 22 tan ttax，那么()taax 33 3 22 sec=+，tdtadx 2 sec=，于是 () c ax ct a dt t t a tdta ta ta ax xdx + + =+= + 22 2 33 3 22 1 cos 1 sec tan1 sec sec tan 3

14、5.+=+= + += + 2222222 2 22 22 222 22 2 2 )ln()ln( 22 ax x caxxaaxx a ax x ax dx adxax ax dxx ()caxx a + 22 2 ln 2 36. ()()() ()c ax x axx ax dx a ax dx dx ax aax ax dxx + + += + + = + + = + 22 22 3 22 2 223 22 222 3 22 2 ln 37. + 22 axx dx 設(shè) = 22 tan ttax，那么()taax 33 3 22 sec=+，tdtadx 2 sec=，于是 c x

15、aax a ctt at dt atata tdta axx dx + + =+= = + 222 22 ln 1 cotcscln 1 sin 1 tansec sec 38. + 222 axx dx 設(shè) = 22 tan ttax，那么()taax 33 3 22 sec=+，tdtadx 2 sec=，于是 c xa xa c ta dt t t atata tdta axx dx + + =+= = + 2 22 2222 2 222 sin 1 sin cos1 sectan sec 39.+dxax 22 設(shè) = 22 tan ttax，那么()taax 33 3 22 sec=

16、+，tdtadx 2 sec=，于是 =+ ttatdttattattdatdtatdtatadxaxtansectansectansectansecsecsecsec 2222232222 () += 1 232222 tanseclnsectansec1secseccttatdtattadttta ()caxx aaxx ctttt a dxax+ + =+=+ 22 2222 22 ln 22 tanseclntansec 2 40.() +dxax 3 22 設(shè) = 22 tan ttax，那么()taax 33 3 22 sec=+，tdtadx 2 sec=，于是 () =+tdt

17、adxax 54 3 22 sec =tdttttttttdtdttansecsectan3tansectansecsec 2335 +=tdttdttttdtttt 353233 sec3sec3tansectansec3tansec () 1 3 tanseclntansec 2 1 secctttttdt+= () 1 535 tanseclntansec 2 3 sec3tansecsecctttttdttttdt+= ()() () a x a axa ctttt a tt a tdtadxax + =+=+ 3 3 224 1 4 3 4 5422 4 tanseclntansec

18、 8 3 tansec 4 sec ()()caxxaaxax x c a axx a xaxa +=+ + + + + 2242222 1 22 2 224 ln 8 3 52 8 ln 8 3 41.dxaxx + 22 設(shè) = 22 tan ttax，那么()taax 33 3 22 sec=+，tdtadx 2 sec=，于是 ()caxc t a t td atdttatdtatatadxaxx+=+=+ 3 22 3 3 4 333222 3 1 cos3cos cos sectansecsectan 42.()()() 2222222 3 222222222222 52 8 ax

19、ax x dxaxadxaxdxaxaaxaxdxaxx+=+=+=+ ()()()()caxx a axax x caxx a ax x aaxxa+=+ + 22 4 222222 2 222224 ln 8 2 8 ln 22 ln 8 3 43.dx x ax + 22 設(shè) = 22 tan ttax，那么()taax 33 3 22 sec=+，tdtadx 2 sec=，于是 lncotcscln cossincos sin cossin sec tan sec 22 22 2 22 axactta t a t dt adt t t a tt dt atdta ta ta dx x

20、 ax +=+=+= + c x aax + + 22 44.dx x ax + 2 22 設(shè) aax的積分 45. 22 ax dx 當(dāng)ax 時(shí)，設(shè) = 2 0sec ttax，那么tataaxtan1sec2 22 =，tdttadxtansec=，于是 ()()caxxc a ax a x ctttdtdt ta tta ax dx +=+ +=+= 22 1 22 22 lnlntanseclnsec tan tansec 當(dāng)ax， 由上段結(jié)果有()() 22 1 22 2222 lnlnaxxcauu au du ax dx +=+= = ()caxxc a axx c xax c+

21、=+ =+ =+ 22 1 2 22 1 22 1 lnln 1 ln 綜上所述，caxx ax dx += 22 22 ln 46. () 3 22 ax dx ，設(shè) = 2 0sec ttax，則()taax 33 3 22 tan=，tdttadxtansec=，于是 () c axa x c ta dt t t a dt t t a dt ta tta ax dx + =+= 222 2222233 3 22sin 1 sin cos1 tan sec1 tan tansec 47. 22 ax xdx ，設(shè) = 2 0sec ttax，則taaxtan 22 =，tdttadxtan

22、sec=，于是 +=+= caxctatdtatdtta ta ta ax xdx 222 22 tansectansec tan sec 48. () 3 22 ax xdx ,設(shè) = 2 0sec ttax，則taaxtan 22 =，tdttadxtansec=，于是 () + =+= c ax ct a dt ta tdtta ta ta ax xdx 22 233 3 22 1 cot 1 sin 11 tansec tan sec 49.caxxaaxxaax x ax dxa dx ax ax ax dxx += + = 22222222 2 22 2 22 22 22 2 ln

23、ln 2 1 2 caxx a ax x += 22 2 22 2 ln 22 50. ()()()() += + = + = 22 2 222 3 22 2 223 22 2 3 22 22 3 22 2 1 ln ax x a aaxx ax dx a ax dx ax dx adx ax ax ax dxx c ax x axxc+ +=+ 22 22 ln 51. 22 axx dx ,設(shè) x時(shí)有 c x a c a t dt tta tta axx dx +=+= arccos tansec tansec 22 當(dāng)0x時(shí)有，c x a a axx dx + = arccos 1 22

24、 ，綜上所述，有c x a a axx dx += arccos 1 22 52. 222 axx dx ，設(shè) = 2 0sec ttax，則taaxtan 22 =，tdttadxtansec=，于是 c xa ax ct at dt a dt tata tta axx dx + =+= = 2 22 2222 222 sin 1 sec 1 tansec tansec 53.dxax 22 ,設(shè) = 2 0sec ttax，則taaxtan 22 =，tdttadxtansec=，于是 += = tt a tt a dt t t atdttatdttatadxaxtansecln 2 ta

25、nsec 2cos cos1 tansectansectan 22 3 2 22222 ()caxx a ax x ctta+=+ 22 2 222 ln 22 tansecln 54.()dxax 3 22 ,設(shè) = 2 0sec ttax，則taaxtan 22 =，tdttadxtansec=，于是 () =tdttatdtttaadxaxsectantansectan 4433 3 22 () +=tdttdttdttdttdtttsecsec2secsec1secsectan 35 2 24 () += 1 35 tanseclntansec 8 3 tansec 4 1 secct

26、ttttttdt () 2 3 tanseclntansec 2 1 secctttttdt+= ;+= 3 tanseclnsecctttdt ()+=+= tttttttttdttdttdttdtttan(sec 2 1 2tanseclntansec 8 3 tansec 4 1 secsec2secsectan 335 4 a ax a x a ax a x a ax a x ccctttt 222222 3 3 321 ln 8 3 8 5 4 1 2tansecln)tansecln + =+ 321 2ccc+ ()()caxxaaxax x caxxaaxx a ax x dx

27、ax+=+= 224222222422 2 22 3 3 22 ln 8 3 52 8 ln 8 3 8 5 4 55.dxaxx 22 ,設(shè) = 2 0sec ttax，則taaxtan 22 =，tdttadxtansec=，于是 = = t dt a t dt adt t t atdttatdttatatadxaxx 2 3 4 3 4 2 322322 coscoscos cos1 sectantansectansec ()()caxct a tatdtatattda+=+=+= 3 223 3 323323 3 1 tan 3 tantantan1tantansec 56.()()

28、22 2 22422222222222222 2 ln 8 3 52 8 axx a axxaaxax x dxaxadxaxaxdxaxx+=+= ()caxx a axax x caxx a +=+ 22 4 222222 4 ln 8 2 8 ln 2 57.dx x ax 22 , 設(shè) x時(shí)，有 ()c x a axcttadttatdtatdtta ta ta dx x ax +=+= arccostan1sectantansec sec tan 2222 22 當(dāng)0x時(shí)，有c x a axdx x ax + = arccos)( 22 22 ；綜上所述，c x a axdx x a

29、x += arccos 22 22 58.dx x ax 2 22 ，設(shè) aax的積分 59.c a x a x dx a xa dx += = arcsin 1 1 222 60. () 3 22 xa dx ,令 = 22 sin ttax，則()taxa 33 3 22 cos=，tdtadxcos=，于是 () + =+= c xaa x ct at dt dt ta ta xa dx 222 2233 3 22 tan 1 coscos cos 61. 22 xa dx ，令 = 22 sin ttax，則taxacos 22 =，tdtadxcos=，于是 += cxatatdta

30、tdta ta ta xa dx 22 22 cossincos cos sin 62. () 3 22 xa xdx ,令 = 22 sin ttax，則()taxa 33 3 22 cos=，tdtadxcos=，于是 () + =+= c xa c ta dt t t a tdta ta ta xa xdx 22 233 3 22 1 cos 1 cos sin1 cos cos sin 63. 22 2 xa dxx ,令 = 22 sin ttax，則taxacos 22 =，tdtadxcos=，于是 cxa x a xa ct a tatdtatdta ta ta xa dxx

31、+=+= 22 22 222 22 22 2 2 arcsin 2 2sin 42 1 sincos cos sin 64. () 3 22 2 xa dxx ,令 = 22 sin ttax，則()taxa 33 3 22 cos=，tdtadxcos=，于是 () c a x xa x cttdt t dt tdta ta ta xa dxx + =+= arcsintan cos cos cos sin 22 233 22 3 22 2 65. 22 xax dx ,令 = 22 sin ttax，則taxacos 22 =，tdtadxcos=，于是 c x xaa a c x xa

32、a x a ctt a dt tta ta xax dx + =+ =+= 2222 2 22 ln 1 ln 1 cotcscln 1 cossin cos 66. 222 xax dx ,令 = 22 sin ttax，則taxacos 22 =，tdtadxcos=，于是 c xa xa ct at dt attaa tdta xax dx + =+= 2 22 22222 222 cot 1 sin 1 cossin cos 67.dxxa 22 ,令 = 22 sin ttax，則taxacos 22 =，tdtadxcos=，于是 cxa x a xa ct a t a dt t

33、atdtatdtatadxxa+=+= + = 22 222 22222 2 arcsin 2 2sin 422 2cos1 coscoscos 68.()dxxa 3 22 ,令 = 22 sin ttax，則()taxa 33 3 22 cos=，tdtadxcos=，于是 () () =+= + = dtt a tdt ata dt t atdtatdtatadxxa2cos 4 2cos 244 2cos1 coscoscos 2 444 2 44433 3 22 () =+ +=+c xaxax xax a a x act ata t a a xa 8 2 2 arcsin 8 3

34、4sin 328 2sin 4 arcsin 4 2222 22 2 4 4444 ()c a x axaxa x +arcsin 8 3 25 8 42222 69.dxxax 22 ,令 = 22 sin ttax，則taxacos 22 =，tdtadxcos=，于是 ()cxact a ttdatdtatatadxxax+=+= 3 223 3 2322 3 1 cos 3 coscoscoscossin 70.dxxax 222 ,令 = 22 sin ttax，則taxacos 22 =，tdtadxcos=，于是 () =tdtatdtadtttatdttatdtatatadxx

35、ax 442422422422222 sinsinsin1sincossincoscossin () + =+= =dt ta t a tdt a t a tat a tadt t adt t a 2 4cos1 44 2cos 4 2sin 44 1 2sin 42 1 4 2cos1 2 2cos1 44 2 44 4 4 4 2 44 ()cxaxa x a xa ct a t a +=+= 2222 444 2 8 arcsin 8 4sin 328 71.dx x xa 22 ,令 = 22 sin ttax，則taxacos 22 =，tdtadxcos=，于是 =+= = cta

36、ttatdta t dt adt t t atdttatdta ta ta dx x xa coscotcsclnsin sinsin sin1 coscotcos sin cos 222 cxa x xaa ac a xa a x xa x a a+ =+ + 22 222222 lnln 72.dx x xa 2 22 ,令 +acbxax的積分 73. + + = + 2 2 22 42 1 a b a c a b x dx a cbxax dx ，令t a b x=+ 2 ，則dtdx = 當(dāng)04 2 acb時(shí)，則令()0 4 4 2 2 2 = uu a acb ，則 = + + =

37、 + 22 2 2 22 1 42 1 ut dt a a b a c a b x dx a cbxax dx 再令rutsec=,rdrrudtsectan=,ruuttan 22 =,于是 += 1 22 tansecln 1 sec 1 tan sectan11 crr a rdr a dr ru rru a ut dt a acb bax a acb a b x u t r 4 2 4 4 2 sec 2 2 2 + = + = ; acb cbxaxa u ut r r r 4 1 2 cos cos1 tan 2 2 222 += = = ccbxaxabax a c acb cb

38、xaxabax a cbxax dx +=+ + = + 2 1 2 2 2 22ln 1 4 22 ln 1 當(dāng)04 2 acb時(shí)，令 a b xt 2 +=，u a acb = 2 4 2 =+dtutadxcbxax 222 ，再令rutsec=，rdrrudttansec= ()crruarrrruardrruadtuta+= tanseclntanseclntansec 2 1 sectan 222222 + + =+= acb cbxaxa acb bax a acb acrruarrua 4 1 2 4 2 2 4 2 1 tansecln 2 1 tansec 2 1 2 2

39、2 2 2 1 22 ccbxaxabax a bac cbxax a bax c acb cbxaxabax a acba + + + =+ + 2 3 2 2 1 2 2 2 2 22ln 8 4 4 2 4 22 ln 4 4 2 當(dāng)04 2 時(shí)，令 2 2 4 4 a bac u = 1 22 22 2222222 ln 22 1 2 1 cutt aa b a ut uta dt a b dt ut t a dt ut a b t a cbxax xdx + = = = + ccbxaxabax aa b a cbxax c a c x a b x a b x aa b a c x

40、a b x a + + =+= 2 2 1 22 22ln 22 ln 2 1 當(dāng)0= u ab ； 2 ba tr +=則 () c ab bax ru dr ba t ba dt + = = + 2 arcsin 24 222 2 82.()() dxxbax；令tax=；則tax+=；dtdx =；于是()() () + =dt ba t ba dxxbax 2 2 24 令0 2 = u ab ； 2 ba tr +=；則()() 4 2 arcsin 22 2 2222 bax c u ru ru r drrudxxbax =+= ()() () c ab baxab dxxbax+

41、 + 2 arcsin 8 2 (十一)含有三角函數(shù)的積分 83.+=cxxdxcossin 84.+=cxxdxsincos 85. +=cxdx x x xdxcosln cos sin tan 86. +=cxdx x x xdxsinln sin cos cot 87.c x x x d xx x d xx xd x dx xdx+ += + + = + + + = + + + = 42 tanln 42 tan 42 tan 42 cos 42 tan 42 2 cos 2 sin2 2 cos sec 2 xx x x x x x x x cotcsc sin cos1 sin 2

42、 sin2 2 cos 2 sin 2 tan 2 = = ; =+ + += cxxxdx 2 cot 2 csclnsec cxx+ tansecln 88.cxxc x x x d xx dx xx dx x dx xdx+=+= cotcscln 2 tanln 2 tan 2 tan 2 cos 2 tan 2 cos 2 sin2 sin csc 2 89.cxxdx+= tansec2 90.cxxdx+= cotcsc2 91.+=cxxdxxsectansec 92.+=cxxdxxcsccotcsc 93.cx x dx x xdx+= = 2sin 4 1 22 2cos

43、1 sin2 94.cx x dx x xdx+= + = 2sin 4 1 22 2cos1 cos2 95.() +=+=xdxxnxxxdxxdxxxxdxdx nnnnnn221111 cossin1sincossincossincoscossinsin ()()()() +=+= xdxnxdxnxxxdxxnxx nnnnn sin1sin1sincossinsin11sincos 21221 +=xdx n n xx n xdx nnn21 sin 1 sincos 1 sin 96.()+=+= xxxdxxnxxxxdxxxxdxdx nnnnnnn1221111 cossi

44、ncossin1cossincossincossinsincoscos ()()()() += xdxnxdxnxxxdxxn nnnn cos1cos1cossincoscos11 2122 +=xdx n n xx n xdx nnn21 cos 1 cossin 1 cos 97. + +=xdx n n xx n xdx n xdx x dx nnnn n 211 sin 1 cossin 1 sin 1 sin sin + = x dx n nx nx dx nnn sin2 1 sin cos 2 1 sin 12 + = x dx n n x x nx dx nnn21 sin1

45、 2 sin cos 1 1 sin 98.() =+= xdxxnxxxxdxxxxdxdx x dx nnnnnn n 221111 sincos1sincoscossinsincossincoscos cos ()() +xdxnxdxnxx nnn cos1cos1sincos 21 + +=xdx n n xx nx dx nn n 21 cos 1 sincos 1 cos 將n換成2n有 + = x dx n n x x nx dx nnn cos2 1 cos sin 2 1 cos 12 + = x dx n n x x nx dx nnn21 cos2 1 cos sin

46、1 1 cos 99. + + + = + =xxd n xx n xxd n xxdxxdxx mnmnnmnmnm1111111 cossin 1 1 cossin 1 1 sincos 1 1 sinsincossincos () + + + = + + + = + dxxxx n m xx n xdxx n m xx n mnnmmnnm22112211 cos1cossin 1 1 sincos 1 1 cossin 1 1 sincos 1 1 + + + + = + xdxx n m xdxx m m xx n nnmnnm cossin 1 1 cossin 1 1 sinco

47、s 1 1 211 + + + + = + +xdxx n m xx n xdxx n m nmnmnm sincos 1 1 sincos 1 1 sincos 1 1 1 211 + + + + =xdxx nm m xx m xdxx nmnmnm sincos 1 sincos 1 1 sincos 211 又有 + + + + = + =x m xx m xxd m xxdxxdxx mmnmnnmnm11111 cos 1 1 cossin 1 1 cossin 1 1 cossincossincos () + + + = + + + = + xxx m n xx m xdxx m

48、 n xx m xd nmmnnmmnn221122111 sin1sincos 1 1 cossin 1 1 sincos 1 1 cossin 1 1 sin + + + + = + xdxx m n xdxx m n xx m dx nmnmmn sincos 1 1 sincos 1 1 cossin 1 1 211 + + + = + + + = + + + 1 1 cossin 1 sincos 1 1 cossin 1 1 sincos 1 1 1 11211 m n xx nm xdxx m n xx m xdxx m n mnnmmnnm xdxx nm2 sincos ；

49、因此 + + + + = + + + =xdxx nm n xx nm xdxx nm m xx nm xdxx nmnmnmnmnm sincos 1 sincos 1 sincos 1 sincos 1 sincos 211211 100.()() () () () () + + =+=cxba ba xba ba dxxbaxbabxdxaxcos 2 1 cos 2 1 sinsin 2 1 cossin 101.()() () () () ()cxba ba xba ba dxxbaxbabxdxax+ + + =+= sin 2 1 sin 2 1 coscos 2 1 sinsi

50、n 102.()() () () () ()cxba ba xba ba dxxbaxbabxdxax+ + + =+= sin 2 1 sin 2 1 coscos 2 1 coscos 103. + + + = + + + = + + = + = + 1 2 tan 2 tan1 1 2 tan 2 tan1 2 tan2 2 tan 2 tan1 2 cos 2 sin2 sin 2 22 2 22 22 2 22 2 2 2 ba a bx a dx x ba a dx ba a bx a ba ax x b x aa dx x xx a dx xba dx + + + = + + =

51、 + + = 2 22 22 22 222 22 222 22 2 22 2 tan 1 2 tan 2 2 tan 2 tan 1 12 2 tan 1 22 sec 2 ba a bx a ba a bx a d a ba ba ax d ba a bx a ba a ba a bx a x d x ba a () 22 2222 2 tan arctan 2 bac ba a bx a ba + + = 104. + = + + = + + = + = + 1 2 tan 2 tan 2 1 2 tan 2 tan1 2 tan2 2 tan 2 tan1 2 cos 2 sin2 si

52、n 22 22 22 2 22 2 2 ab a bx a x d ab a dx ab a bx a x ab a dx x b x aa x xx ba dx xba dx 令t ab b x a sec 2 tan 22 = + ；原式 222222 2 22 22 2 csc 2 tan sec2 tan sectan 2 ab tdt ab dt t t ab t tdtt a ab ab a = = = = () 22 22 22 2222 2 tan 2 tan ln 1 2 tanln 2 cotcsclnbac abb x a abb x a ab c x ab ctt+ +

53、 + + = 106. () ()()() + + = + = + + = + = + 2 tan1 2 tan 2 2 tan 2 sec 2 tan 2 tan1 2 sin 2 cos cos22 2 2 2 22 x ba ab x d ba x abba dx x dx x baba x xx ba dx xba dx 令 = + 2 0sin 2 tan uu x ba ab ；則udu ab bax dcos 2 tan + =；于是原式 c u ab ba ba cuu ab ba ba udu ab ba bau udu ab ba ba + + + + =+ + + = + + = + + = 24 tanln 2 tan