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1、常常 用用 积积 分分 公公 式式一含有一含有axb的积分的积分a 01dx1ln ax b Cax ba1(ax b)1C 1a(1)2(ax b)dx3x1dx2(ax bbln ax b)Cax bax21 1dx3(ax b)22b(ax b)b2ln ax bC4a2ax b5671ax bdxClnbxx(ax b)1aax bdxlnC22bxbxx(ax b)x1bdx(ln ax b)C(ax b)2a2ax bx21b2dx3(ax b2bln ax b)C82(ax b)aax b911ax bdxlnCb(ax b)b2xx(ax b)2二含有二含有ax b的积分的积分
2、2(ax b)3C3a2311x ax bdx(3ax 2b)(ax b)C215a2(15a2x212abx 8b2)(ax b)3C12x2ax bdx3105a10ax bdx1314x2dx2(ax 2b)ax b C3aax bx22dx(3a2x24abx 8b2)ax b C315aax bdx15x ax b1lnbax b bC(b 0)ax b b2ax barctanC(b 0)bb161718x2dxax badxbx2bx ax bax bax bdxdx2 ax b bxx ax bax bax badxdx2xx2x ax b三含有三含有x2 a2的积分的积分19
3、2021dx1x=arctanCx2a2aadxx2n 3dx=(x2 a2)n2(n 1)a2(x2a2)n12(n 1)a2(x2 a2)n11x adxlnC=22x a2ax a四含有四含有ax2b(a 0)的积分的积分1arctandxab222ax b1ln2 abax Cb(b 0)ax bC(b 0)ax b23x12dxln ax b C2ax b2ax2xbdx242dx2ax baaax bdx1x2C25ln2x(ax2b)2bax b26dx1adxx2(ax2b)bxbax2b27dxaax2b1x3(ax2b)2b2lnx22bx2C28dxx1dx(ax2b)2
4、2b(ax2b)2bax2b五含有五含有ax2bx c(a 0)的积分的积分22ax 29dx4ac b2arctanb4ac b2Cax2bx c12ax bb24acb24acln2ax bb24acC30 x1bdxax2bx cdx2aln ax2bx c 2aax2bx c六含有六含有x2a2(a 0)的积分的积分31dxxx2 a2arshaC1ln(x x2a2)C32dx(x2a2)3xa2x2a2C33xx2 a2dxx2a2C34x(x2a2)3dx1x2 a2C35x2xx2a2dx2x2 a a222ln(x x2 a2)C36x2(x2a2)3dxxx2 a2ln(x
5、 x2 a2)Cdx1x2a237ax x2 a2alnxC38dxx2 a2x2x2 a2a2xC39x2a2dxx2x2 a2a22ln(x x2 a2)C(b2 4ac)(b2 4ac)40(x2a2)3dx(2x25a2)x2a2a4ln(x x2a2)C41x x2a2dx42x4344222x8381(x2a2)3C3xa42222x a dx(2x a)x a ln(x x2a2)C88x2a2x2a2a22dxx a alnCxxx2 a2x2 a222dxln(x x a)C2xx七含有七含有x2a2(a 0)的积分的积分454647484950515253dxx2a2xxa
6、rchC1=ln x x2a2Cxaxa2dx(x a)xx2a2223x a22Cdxx2a2Cx(x a)22 3dx1x a22Cx2a22x a ln x x2a2Cdx22x2a2x2x2(x2a2)3dxx x2a2dx1axx2a2ln x x2a2CarccosaCxx22x2a2C222a xx adxx2a22x a ln x x2a2Cx a dx22254(x2a2)3dx(2x25a2)x2a2a4ln x x2a2Cx83855x x2a2dx56x57582221(x2a2)3C3xa42222x a dx(2x a)x a ln x x2a2C88ax2a2dx
7、x2a2aarccosCxxx2a2x2a222dxln x x aC2xx八含有八含有a2 x2(a 0)的积分的积分596061626364dxa2 x2arcsinC3xa2xadx(a x)xa2 x222a x22Cdx a2 x2Cx(a x)22 3dx1a x22Cx2a2x2a x arcsinCdx22aa2 x2x2x2(a2 x2)3dxdxxa2 x2arcsinxCa1a a2 x2C65ln22axx a x6667x22a2 x2C222a xa xdxx2a2x2a x arcsinCa x dx22a2x8169x a2 x2dx(a2 x2)3C368(a
8、2 x2)3dx(5a22x2)a2 x2a4arcsinC38xa70 x71722xa4x2222a x dx(2x a)a x arcsinC88a22a2 x2a a2 x222dxa x alnCxxa2 x2a2 x2xdxarcsinCx2xa九含有九含有ax2bxc(a 0)的积分的积分73dxax2bx c1ln 2ax b2 aax2bx c Ca74ax2bx cdx75767778xax2bx cdxc bx ax222ax bax2bx c4a1ax2bx ca12ax barcsinC2ab 4acdx2ax bb24ac2ax b2c bx ax dxcbx ax
9、 arcsinC324a8 ab 4acxc bx ax2dx1b2ax bc bx ax2arcsinC32a2 ab 4ac十含有十含有x a或或(x a)(b x)的积分的积分x bx b)C79808182x ax adx(x b)(ba)ln(x a x bx bx ax ax adx(x b)(ba)arcsinCb xb xb xx adxC2arcsinb x(x a)(b x)(a b)2x a b(ba)2x a(x a)(b x)arcsinC(x a)(b x)dx44b x十一含有三角函数的积分十一含有三角函数的积分83sin xdxcosx C84cosxdxsin
10、x C85tan xdxln cosx C86cot xdxln sinx C87sec xdxln tan()Cln secx tanx C88csc xdxln tanxCln cscxcotx C24x289sec2xdxtan x C90csc2xdxcot x C91secxtan xdxsecx C92cscxcot xdxcscx Cx124x194cos2xdxsin2x C241n 1n295sinnxdxsinn1xcosx sinxdxnn1n 1n296cosnxdxcosn1xsinx cosxdxnndx1cosxn 2dx97nn1sin xn 1 sinxn 1
11、sinn2xdx1sin xn 2dx98nn1n2cos xn 1 cosxn 1cosx1m1m2n99cosmxsinnxdxcosm1xsinn1x cosxsin xdxm nm n1n 1cosm1xsinn1x cosmxsinn2xdxm nm n93sin2xdxsin2x C100sinaxcosbxdx101sinaxsinbxdx11cos(a b)x cos(a b)x C2(a b)2(a b)11sin(a b)x sin(a b)x C2(a b)2(a b)102cosaxcosbxdx11sin(a b)x sin(a b)x C2(a b)2(a b)10
12、32dx22arctana bsin xa batanxb2C22a b(a2 b2)x22bb adx1210422lnCxa bsin x22b aatanbb a2atan(a2 b2)1052a ba bxdxarctan(tan)Ca ba ba b2a bcosx(a2 b2)xdx1a b2106lna bcosxa bbaxtan2tana bbaCa bba(a2 b2)107108dx1barctan(tan x)C2222a cos x b sin xaba1btan x adxlnC2abbtan x aa2cos2x b2sin2x11sinax xcosax C2a
13、a122110 x2sinaxdxx2cosax 2xsinax 3cosax Caaa11111xcosaxdx2cosax xsinax Caa122112x2cosaxdxx2sinax 2xcosax 3sinax Caaa109xsinaxdx十二含有反三角函数的积分其中a 0113arcsin dxxarcsina2 x2Cx2a2xx2xa x2C114xarcsindx()arcsin24a4ax3x1x115x arcsin dxarcsin(x2 2a2)a2 x2C3a9a2xaxa116arccos dxxarccosa2 x2Cxaxax2a2xx2xa x2C117
14、xarccos dx()arccos24a4ax3x1x118x arccosdxarccos(x22a2)a2 x2C3a9a2119arctandxxarctanln(a2 x2)C120 xarctandx(a2 x2)arctanx Cx3xa2a3x121x arctandxarctanx ln(a2 x2)C3a66a2xaxaa2xa12xaa2十三含有指数函数的积分十三含有指数函数的积分1xa Clna1123eaxdxeaxCa1124xeaxdx2(ax 1)eaxCa1n125xneaxdxxneaxxn1eaxdxaa122axdx126xaxdxxx1xa a C2l
15、na(lna)1nxnn1xx a xa dxlnalna1128eaxsinbxdx22eax(asinbx bcosbx)Ca b1129eaxcosbxdx22eax(bsinbx acosbx)Ca b1130eaxsinnbxdx222eaxsinn1bx(asinbx nbcosbx)a b n1131eaxcosnbxdx222eaxcosn1bx(acosbx nbsinbx)a b n127xnaxdx十四含有对数函数的积分十四含有对数函数的积分132ln xdxxlnx x Cdxln lnx Cxln x1n11x(ln x)C134xnln xdxn 1n 113313
16、5(ln x)ndxx(ln x)nn(ln x)dx136xm(ln x)ndx1nxm1(ln x)nxm(ln x)n1dxm1m1n1十五含有双曲函数的积分十五含有双曲函数的积分137shxdxchx C138chxdxshx C139thxdxlnchx C140sh2xdxsh2x C141ch2xdxsh2x C十六定积分十六定积分142cosnxdxsinnxdx0143cosmxsinnxdx00,m ncosmxcosnxdx144,m nx214x214145sinmxsinnxdx0,m n,m n1460sinmxsinnxdx020n0,m ncosmxcosnxdx,m n 2147Insin xdxcosnxdxn 1In2nn 1 n 3Innn 2n1 n3Innn220In4 2n为大于 1 的正奇数,I115 33 1 n为正偶数,I04 2 22