61．?xa?xx22dx＝?a?x?C
2262．?(a?x)223dx＝1a?x22?C
63．?x222dx＝?x2a?x?22a2a?x2arcsinxa?C
64．?x2223dx＝xa?x22?arcsinxa?C
(a?x)dxxa?xdxx22265．?＝
1alna?a?xx22?C
66．?a?x2222＝?a?xax2222?C
67．?68．?a?xdx＝x2a?x?x81322a222arcsin2xa2?C
384(a?x)dx＝
22223(5a?2x)a?x?(a?x)?C
223aarcsinxa?C
69．?xa?xdx＝?70．?x2a?xdx＝
22x82(2x?a)a?x?2222a482arcsinxa?C
71．?a?xxa?xx2222dx＝a?x?aln2a?a?xxxa2?C
272．?dx＝?a?xx22?arcsin?C
（九）含有?ax?bx?c(a?0)的积分 73．?dxax?bx?c22＝1aln2ax?b?2aax?bx?c?C
274．?
ax?bx?cdx＝22ax?b4aax?bx?c 22?4ac?b8a3ln2ax?b?2aax?2bx?c? C75．?xax?bx?c2dx＝1aax?bx?c
2 ?b2a3ln2ax?b?2aax?2bx?c? C76．?dxc?bx?ax2＝?1aarcsin2ax?bb?4ac22?C
77．?c?bx?axdx＝22ax?b4ac?bx?ax?b?4ac8a32arcsin2ax?bb?4ac2?C
78．?xc?bx?ax2dx＝?1ac?bx?ax?2b2a3arcsin2ax?bb?4ac2?C
（十）含有?x?ax?b或(x?a)(b?x)的积分
79．?x?ax?bx?ab?xdx＝(x?b)x?ax?bx?ab?x?(b?a)ln(x?a?x?b)?C
80．?dx＝(x?b)?(b?a)arcsinx?ab?x?C
81．?dx(x?a)(b?x)＝2arcsinx?ab?x?C(a?b)
82．?(x?a)(b?x)dx＝2x?a?b4(x?a)(b?x)?(b?a)42arcsinx?ab?x?C
(a?b) （十一）含有三角函数的积分 83．?sinxdx＝?cosx?C
84．?cosxdx＝sinx?C 85．?tanxdx＝?lncosx?C 86．?cotxdx＝lnsinx?C 87．?secxdx＝lntan(?4x2?x2)?C＝lnsecx?tanx?C
88．?cscxdx＝lntan?C＝lncscx?cotx?C
289．?secxdx＝tanx?C 290．?cscxdx＝?cotx?C
91．?secxtanxdx＝secx?C 92．?cscxcotxdx＝?cscx?C 93．?sinxdx＝94．?cosxdx＝
n22x2x2??1n1414sin2x?C sin2x?C
n?195．?sinxdx＝?96．?cosxdx＝97．?98．?dxsinxdxcosxmnnsinxcosx?n?1n1ncos1?n?1xsinx??nn?1n?sinn?2xdx
?cosn?2xdx
＝?＝
cosxn?1n?2n?1n?1m?1n?1sin1?x?sindxn?2xx
m?1m?nsinxn?1n?1cosnx?n?2?cosdxn?299．?cosxsinxdx＝
1m?n1cosxsinn?1x?n?1?cosm?2xsinxdx xsinn?2n＝?m?ncosm?1xsinx??cosm?n1n?1mxdx
100．?sinaxcosbxdx＝?12(a?b)cos(a?b)x?2(a?b)cos(a?b)x?C
101．?sinaxsinbxdx＝?12(a?b)1sin(a?b)x?12(a?b)1sin(a?b)x?C
102．?cosaxcosbxdx＝
2(a?b)sin(a?b)x?2(a?b)sin(a?b)x?C
103．?dxa?bsinx＝
2a?b22atanarctan2x2?b2?C(a?b)
22a?b104．?dxa?bsinx＝
21b?a2atanlnatanx2x2?b??b?b?ab?ax2222?C2(a?b)
22105．?dxa?bcosx＝2a?ba?ba?barctan(a?ba?btan)?C(a?b)
22106．?dxa?bcosx＝1a?ba?bb?atanlntanx2x2ba??a?bb?aa?bb?a?C(a?b)
22107．?108．?dxacosx?bsinxdxacosx?bsinx1a222222222＝
1abarctan(tanx)?C
＝
12ab1alnbtanx?abtanx?a?C
109．?xsinaxdx＝
2sinax?1a2xcosax?C
2a2110．?xsinaxdx＝?111．?xcosaxdx＝
2xcosax?1axsinax?2a3cosax?C
1a2cosax?2xsinax?C 2a2112．?xcosaxdx＝
x1axsinax?xcosax?2a3sinax?C
（十二）含有反三角函数的积分（其中a?0） 113．?arcsindx＝xarcsinaxa?a?x?C
22114．?xarcsinxadx＝(x223?a24)arcsinxa?x422a?x?C
115．?xarcsindx＝
ax2xx3arcsinxa2xa?192(x?2a)a?x?C
22222116．?arccosdx＝xarccosa?a?x?C
117．?xarccosdx＝(axaxx22x3?a4)arccosxa?x4a?x?C
22118．?xarccos119．?arctanxa2dx＝
3arccosxaxaa2?19(x?2a)a?x?C
2222dx＝xarctanxadx＝xa122?2ln(a?x)?C
xa222120．?xarctan2(a?x)arctan?a2x?C
121．?xarctandx＝
x33arctanxa?a6x?a36ln(a?x)?C
22（十三）含有指数函数的积分 122．?adx＝123．?edx＝
axx1lna1ae1a2a?C
axxax?C
ax124．?xedx＝
nax(ax?1)exenax?C
n?1125．?xedx＝126．?xadx＝
nx1a??xa12nedx
axxxlna1a?x(lna)xa?C
x127．?xadx＝
axlnaxa?1a?b1222nnlnaax?xn?1adx
x128．?esinbxdx＝129．?ecosbxdx＝
axe(asinbx?bcosbx)?C e(bsinbx?acosbx)?C
axa?b2
高等数学积分公式大全
