高等数学积分公式大全.pdf

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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 0)1dx1ax baln ax b C2(ax b)dx1(ax b)1C（1）a(1)3x1dxax ba2(ax bbln ax b)Cx21 1dx3(ax b)22b(ax b)b2ln ax bC4a2ax b51ax bdxx(ax b)blnxC61aax bdxlnC22bxbxx(ax b)7x1bdx(ln ax b)C(ax b)2a2ax bx21b2dx3(ax b2bln ax b)C82(ax b)aax b911ax bdxlnCxx(ax b)2b(ax b)b2（二）含有（二）含

2、有ax b的积分的积分23(ax b)C3a23(3ax 2b)(ax b)C11x ax bdx215a22(15a2x212abx 8b2)(ax b)3C12xax bdx3105a10ax bdx13x2dx2(ax 2b)ax b C3aax b14x22dx(3a2x24abx 8b2)ax b C315aax bdx15x ax b161lnbax b bC(b 0)ax b b2ax barctanC(b 0)bbx2dxax badxbx2bx ax bax b17ax bdxdx2 ax b bxx ax bax bax badxdxx2x2x ax b2218（三）含有（

3、三）含有x a的积分的积分19dx1x=arctanCx2a2aadxx2n 3dx=(x2 a2)n2(n 1)a2(x2a2)n12(n 1)a2(x2 a2)n11x adxln=x2a22ax aC22021（四）含有（四）含有ax b(a 0)的积分的积分1arctandxab222ax b1ln2 ab23ax Cb(b 0)ax bC(b 0)ax bx12dxax2b2aln ax b C24x2ax2bdxxabadxax2bdxx(ax2b)1x2252blnax2bC26dx1ax2(ax2b)bxdxbax2b27dxaax2b1x3(ax2b)2b2lnx22bx2C

4、28dxx(ax2b)22b(ax2b)1dx2bax2b（五）含有（五）含有ax2bx c(a 0)的积分的积分22ax2arctanb29dx4ac b4ac b2Cax2bx c12ax bb2ln4acCb24ac2ax bb24ac30 x1bdxax2bx cdx2aln ax2bx c 2aax2bx c（六）含有（六）含有x2a2(a 0)的积分的积分31dxx2 a2arshxaC1ln(x x2a2)C32dxx(x2a2)3a2x2 a2C33xx2 a2dxx2a2C34x1(x2a2)3dxx2 a2C(b2 4ac)(b2 4ac)35x2a22x a ln(x x

5、2 a2)Cdx22x2a2x236x2(x2 a2)3dxdxxx2 a2ln(x x2 a2)C1x2a2aC37ln22axx x a38x2x2 a2C222a xx adx39x2a22x a ln(x x2 a2)Cx a dx2222x342232222(x a)dx(2x 5a)x a a ln(x x2a2)C8812241x x a dx(x2a2)3C34042x2xa42222x a dx(2x a)x a ln(x x2a2)C882243x2a2x2a2a22dxx a alnCxxx2a2x2 a2dxln(x x2 a2)C2xx44（七）含有（七）含有x2a2

6、(a 0)的积分的积分45dxx2a2xxarchC1=ln x x2a2Cxa46dx(x a)xx2a222 3xa2x a22C47dxx2a2C48x(x a)22 3dx1x a22C49xxx2a22x a ln x x2a2Cdx22x2a2x250 x2(x2a2)3dxx2a2dxdxxx2a2ln x x2a2C511aarccosCax522x2a2C222a xx a2253x2a22x a ln x x2a2Cx a dx22x34223222222(x a)dx(2x 5a)x a a ln x x aC8812255x x a dx(x2a2)3C35456x2x

7、a42222x a dx(2x a)x a ln x x2a2C882257ax2a2dxx2a2aarccosCxxx2a2x2a222dxln x x aC2xx58（八）含有（八）含有a2 x2(a 0)的积分的积分59dxa2 x2arcsinxCaxC60dx(a x)22 3a2a x2261xa2 x2dx a2 x2C1a x2262x(a x)22 3dxC63x2a2x2a x arcsinCdx22aa2 x2x264x2(a2 x2)3dxdxxa2 x2arcsinxCa1a a2 x2C65ln22axx a x66x2a2 x2C222a xa xdx67x2a2

8、x2a x arcsinCa x dx22a22x34x2232222(a x)dx(5a 2x)a x a arcsinC88a12269x a x dx(a2 x2)3C36870 x2xa4x2222a x dx(2x a)a x arcsinC88a2271a2 x2a a2 x222dxa x alnCxxa2 x2a2 x2xdxarcsinCx2xa72（九）含有（九）含有ax2bxc(a 0)的积分的积分73dxax2bx c1ln 2ax b2 aax2bx c Ca74ax2bx cdx2ax bax2bx c4axax2bx c4acb28 a3ln 2ax b2 a a

9、x2bx c C75dx1ax2bx cab2 a3ln 2ax b 2 aax2bx c Cdxc bx ax227612ax barcsinC2ab 4ac772ax bb24ac2ax b2c bx ax dxcbx ax arcsinC324a8 ab 4acxc bx ax2dx1b2ax bc bx ax2arcsinC32a2 ab 4ac78（十）含有（十）含有x a或或(x a)(b x)的积分的积分x bx b)C79x ax adx(x b)(ba)ln(x a x bx b80 x ax ax adx(x b)(ba)arcsinCb xb xb xx adxC2arc

10、sinb x(x a)(b x)81(a b)822x a b(ba)2x a(x a)(b x)arcsinC(x a)(b x)dx44b x(a b)（十一）含有三角函数的积分（十一）含有三角函数的积分83sin xdxcosx C84cosxdxsinx C85tan xdxln cosx C86cot xdxln sinx C87sec xdxln tan(x)Cln secx tanx C4288csc xdxln tanxCln cscxcotx C289sec xdxtanx C90csc xdxcotx C91secxtan xdxsecx C92cscxcot xdxcsc

11、x C22x1sin2x C24x1294cos xdxsin2x C241n1n 1nn295sin xdxsinxcosx sinxdxnn1n 1nn196cos xdxcosxsin x cosn2xdxnndx1cosxn 2dx97nn1n2sin xn 1 sinxn 1sinxdx1sin xn 2dx98nn1n2cos xn 1 cosxn 1cosx1m 1mncosm1xsinn1x cosm2xsinnxdx99cos xsin xdxm nm n1n 1cosm1xsinn1x cosmxsinn2xdxm nm n93sin xdx2100sinaxcosbxdx

12、11cos(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)101sinaxsinbxdx102cosaxcosbxdx11sin(a b)x sin(a b)x C2(a b)2(a b)1032dxa bsin xa2b2arctanatanxb2C22a b(a2 b2)x22bb adx12104lnC22xa bsin x22b aatanbb a2atan105(a2 b2)2a ba bxdxarctan(tan)Ca bcosxa ba ba b2(a2 b2)xdx1a b2106lna bcos

13、xa bbaxtan2tan107a bbaCa bba(a2 b2)dx1barctan(tan x)Ca2cos2x b2sin2xaba1btan x adxlna2cos2x b2sin2x2abbtan x aC10811sinax xcosax C2aa12222110 x sinaxdxx cosax 2xsinax 3cosax Caaa11111xcosaxdx2cosax xsinax Caa12222112x cosaxdxx sinax 2xcosax 3sinax Caaa109xsinaxdx（十二）含有反三角函数的积分（其中（十二）含有反三角函数的积分（其中a 0

14、）113arcsin dxxarcsinxaxa2 x2Cax2a2xx2xa x2C114xarcsindx()arcsin24a4ax3x12x222115x arcsin dxarcsin(x 2a)a x C3a9a2116arccos dxxarccosxaxa2 x2Cax2a2xx2x)arccosa x2C117xarccos dx(24a4ax3x12x222118x arccosdxarccos(x 2a)a x C3a9a2xxadxxarctanln(a2 x2)Caa2x12xa2120 xarctandx(a x)arctanx Ca2a2119arctanx3xa

15、2a3xln(a2 x2)C121x arctandxarctanx 3a66a2（十三）含有指数函数的积分（十三）含有指数函数的积分1xa Clna1axax123e dxeCa1axax124xe dx2(ax 1)eCa1naxnn1 axnax125x e dxx exe dxaa122a dxx126xa dxnxxxx1a axC2lna(lna)1nxnx a xn1axdxlnalna1axaxe(asinbx bcosbx)C128e sinbxdx22a b127x a dx1axe(bsinbx acosbx)C22a b1axn130e sin bxdx2eaxsinn

16、1bx(asinbx nbcosbx)22a b n129e cosbxdxaxn(n 1)b2axn2e sinbxdx2a b2n2131e cos bxdxaxn1eaxcosn1bx(acosbx nbsinbx)222a b nn(n 1)b2axecosn2bxdx222a b n（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 Cdxxln xln lnx C1n11n134x ln xdxx(ln x)Cn 1n 1133n135(ln x)dxx(ln x)n(ln x)nn1dx136x(ln x)dxmn1nxm1(ln x)nxm(ln

17、 x)n1dxm1m1（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 Cx1sh2x C24x12141ch xdxsh2x C24140sh xdx2（十六）定积分（十六）定积分142cosnxdxsinnxdx0143cosmxsinnxdx01440,m ncosmxcosnxdx,m n1450,m nsinmxsinnxdx,m n1460sinmxsinnxdx2000,m ncosmxcosnxdx,m n 2147InInsin xdxcosnxdxn20n 1In2nn 1 n 3Innn 2n1 n3Innn24 2，I11（n为大于 1 的正奇数）5 33 1，I0（n为正偶数）4 2 22