高等数学常用积分公式查询表.pdf

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1、.-导数公式：导数公式：(tgx)sec x(ctgx)csc2x(secx)secxtgx(cscx)cscxctgx(ax)axlna(logax)基本积分表：基本积分表：2(arcsin x)11xlna1 x21(arccosx)1 x21(arctgx)1 x21(arcctgx)1 x2tgxdx lncosx Cctgxdx lnsinx Csecxdx lnsecxtgx Ccscxdx lncscxctgx Cdx1xarctgCa2 x2aadx1xalnx2a22axaCdx1a xa2 x22alna xCdxx arcsinCa2 x2a2ndx2 seccos2xx

2、dx tgxCdx2 cscsin2xxdx ctgxCsecxtgxdx secxCcscxctgxdx cscxCaxa dx lnaCxshxdx chxCchxdx shxCdxx2a2 ln(xx2a2)C2Insin xdx cosnxdx 00n1In2nx2a22x a dx x a ln(xx2a2)C22x2a2222x a dx x a ln xx2a2C22x2a2x222a x dx a x arcsinC22a22三角函数的有理式积分：三角函数的有理式积分：-可修编-.-sin x 2u1u2，cosx 1u2x2du1u2，u tg2，dx 1u2（一）含有（一）

3、含有ax b的积分的积分(a 0)1dxax b1aln ax b C2(axb)dx1a(1)(ax b)1C（1）3x1ax bdxa2(ax bbln ax b)C4x2ax bdx1 1a2(ax b)22b(ax b)b2ln ax b3C5dxx(ax b)1blnax bxC6dxx2(ax b)1bxab2lnax bxC7x(ax b)2dx1a2(ln ax b bax b)C8x21b2(ax b)2dxa3(ax b2bln ax b ax b)C9dxx(ax b)21b(ax b)1b2lnax bxC（二）含有（二）含有ax b的积分的积分10ax bdx23a(

4、ax b)3C11x ax bdx2(3ax 2b)(ax b)315a2C12x2ax bdx2105a3(15a2x212abx 8b2)(ax b)3C13x2ax bdx3a2(ax 2b)ax b C-可修编-.-14x2ax bdx215a3(3a2x24abx 8b2)ax b C1ax b b15dxblnax b bC(b 0)x ax b2barctanax bbC(b 0)16dxax bx2ax bbxadx2bx ax b17ax bxdx2 ax b bdxx ax b18ax bax bx2dxadxx2x ax b（三）含有（三）含有x2 a2的积分的积分19d

5、xx2a2=1aarctanxaC20dxx2n 3dx(x2 a2)n=2(n 1)a2(x2a2)n12(n 1)a2(x2 a2)n121dx1x ax2a2=2alnx aC（四）含有（四）含有ax2b(a 0)的积分的积分1a22dxarctanabbx C(b 0)ax2b1ax b2 ablnax bC(b 0)23x1ax2bdx22aln ax b C-可修编-.-24x2ax2bdxxbdxaaax2b25dx1x2x(ax2b)2blnax2bC26dxx2(ax2b)1bxadxbax2b27dxaax2b1x3(ax2b)2b2lnx22bx2C28dx(ax2b)2

6、x1d2b(ax2b)x2bax2b（五）含有（五）含有ax2bx c(a 0)的积分的积分22ax dx4ac b2arctanb4ac b2C29ax2bx c12ax bb24b24aclnac2ax bb24acC30 xax2bx cdx12aln ax2bx c bdx2aax2bx c（六）含有（六）含有x2a2(a 0)的积分的积分31dxxx2 a2arshaC1ln(x x2a2)C32dxx(x2a2)3a2x2 a2C33xx2 a2dxx2a2C34x1(x2a2)3dxx2 a2C-(b2 4ac)(b2 4ac)可修编-.-35x2x2a2dxx2x2a2a22l

7、n(xx2a2)C36x2(x2a2)3dxxx2a2ln(xx2a2)C37dx1x2a2ax x2a2alnxC38dxx2x2a2x2a2a2xC39x2a2dxx2x2a2a22ln(xx2a2)C40(x2a2)3dxx8(2x25a2)x2a238a4ln(xx2a2)C41x x2a2dx1(x2a23)3C42x2x2a2dxx8(2xa2)x2a2a428ln(xx2a2)C243xa2x2xdxx2a2alna2axC44x2a2x2a2x2dxxln(xx2a2)C（七）含有（七）含有x2a2(a0)的积分的积分45dxxxarchxx2a2aC1=lnxx2a2C46d

8、xx(x2a2)3a2x2a2C47xx2a2dxx2a2C-可修编-.-48x(x2a2)3dx1x2a2C49x22a2dxx2x2x2a2a2ln x x2a2C50 x2(x2a2)3dxxx2a2ln x x2a2C51dxx x2a21aarccosaxC52dxx2a2x2x2a2a2xC253x2a2dxx2a2x a22ln x x2a2C54(x2a2)3dxx348(2x25a2)x2a28a ln x x2a2C55x x2a2dx13(x2a2)3C56x2x2a2dxx8(2x a2)x2a2a428ln x x2a2C57x2a2xdxx2a2aarccosaxC

9、58x2a2x2x2dxa222xln x x aC（八）含有（八）含有a2 x2(a 0)的积分的积分59dxxa2 x2arcsinaC60dxx(a2 x2)3a2a2 x2C-可修编-.-61x2a2 x2dx a x2C62x1(a2 x2)3dxa2 x2C63x2xa2 x2dx2a2 x2a2x2arcsinaC264x(a2 x2x)3dxa2 x2arcsinxaC65dx1a a2x a2 x2alnx2xC66dxa2 x2x2a2 x2a2xC67a2 x2dxx22a2 x2a2arcsinxaC68(a2 x2)3dxx222234x8(5a 2x)a x 8a

10、arcsinaC69x a2 x2dx13(a2 x2)3C70 x2a2 x2dxx2a4x8(2x a2)a2 x28arcsinaC71a2 x222a a2 x2xdxa x alnxCa272 x2x2dxa2 x2xarcsinxaC（九）含有（九）含有ax2bxc(a 0)的积分的积分73dx1ax2bx caln 2ax b2 aax2bx c C-可修编-.-74ax2bx cdx2ax bax2bx c4axax bx c24acb28 a3ln 2ax b2 a ax2bx c C75dx1ax2bx cabdxc bx ax222 a3ln 2ax b 2 aax2bx

11、 c C7612ax barcsinC2ab 4ac772ax bb24ac2ax b2c bx ax dxcbx ax arcsinC324a8 ab 4acxc bx ax2x ax b78dx1b2ax bc bx ax2arcsinC32a2 ab 4ac（十）含有（十）含有或或(x a)(b x)的积分的积分79x ax adx(x b)(ba)ln(x a x bx bx b)C80 x ax ax adx(x b)(ba)arcsinCb xb xb xdx(x a)(b x)2arcsin81x aCb x(a b)822x a b(ba)2x a(x a)(b x)arcsi

12、nC(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-可修编-.-83sin xdxcosx C84cosxdxsinx C85tan xdxln cosx C86cot xdxln sinx C87sec xdxln tan(4x2)Cln secx tanx C88csc xdxln tanx2Cln cscxcotx C89sec2xdxtanx C90csc2xdxcotx C91secxtan xdxsecx C92cscxcot xdxcscx C93sin2xdxx214sin2x C94cos2xdxx214sin2x C95si

13、nnxdx1n1n 1n2nsinxcosx nsinxdx96cosnxdx1n1n 1ncosxsin x ncosn2xdx97dx1cosxn 2dxsinnxn 1sinn1xn 1sinn2x98dx1sin xn 2dxcosnxn 1cosn1xn 1cosn2x99cosmxsinnxdx1m ncosm1xsinn1x m 1m ncosm2xsinnxdx1cosm1xsinn1x n 1cosmm nm nxsinn2xdx-可修编-.-100sinaxcosbxdx11cos(a b)x cos(a b)x C2(a b)2(a b)101sinaxsinbxdx12

14、(a b)sin(a b)x 12(a b)sin(a b)x C102cosaxcosbxdx12(a b)sin(a b)x 12(a b)sin(a b)x Catanx103dx22b2a bsin xa2b2arctana2b2C(a2 b)dxatanx1221042bb aa bsin xb2a2ln(a2 b2)atanxC2bb2a2105dx2a ba bx2a bcosxa ba barctan(a btan2)C(a b2)tanxa b106dx1a b2baa bcosxa bbaln(a2 b2)tanxa bC2ba107dxa2cos2x b2sin2x1ab

15、arctan(batan x)C108dx1btan x a2cos2x b2sin2x2ablnabtan x aC109xsinaxdx1a2sinax 1axcosax C110 x2sinaxdx1222ax cosax a2xsinax a3cosax C111xcosaxdx11a2cosax axsinax C112x2cosaxdx1ax2sinax 22a2xcosax a3sinax C（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）-可修编-.-113arcsinxadxxarcsinxaa2 x2C114xarcsinxx2adx(2a24)

16、arcsinxx2a4a x2C115x2arcsinxx3adx3arcsinxa19(x2 2a2)a2 x2C116arccosxadxxarccosxaa2 x2C117xarccosxx2adx(2a24)arccosxax4a2 x2C118x2arccosxx3adxarccosx1(x22a2)a23a9 x2C119arctanxdxxarctanxaln(a2aa2 x2)C120 xarctanxadx12(a2 x2)arctanxaa2x C121x2arctanxx3xa3adx3arctana6x2a6ln(a2 x2)C（十三）含有指数函数的积分（十三）含有指数

17、函数的积分122axdx1lnaaxC123eaxdx1axaeC124xeaxdx1axa2(ax 1)eC125xneaxdx1naxnnax eax1eaxdx126xaxdxxlnaax1(lna)2axC127xnaxdx1lnaxnaxnlnaxn1axdx128eaxsinbxdx1a2b2eax(asinbx bcosbx)C-可修编-.-129eaxcosbxdx1a2b2eax(bsinbx acosbx)C130eaxsinnbxdx1axn1a2b2n2e sinbx(asinbx nbcosbx)n(n 1)b2axn2a2b2n2e sinbxdx131eaxcos

18、nbxdx1a2b2n2eaxcosn1bx(acosbx nbsinbx)n(n 1)b2axa2b2n2ecosn2bxdx（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 C133dxxln xln lnx C134xnlnxdx1n 1xn1(ln x 1n 1)C135(lnx)ndxx(ln x)nn(ln x)n1dx136xm(lnx)ndx1xm1(ln x)nnm1xm(ln x)n1m1dx（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 C140sh2xdxx214sh2x C141ch2xdxx124sh2x C（十六）定积分（十六）定积分142cosnxdxsinnxdx0143cosmxsinnxdx0-可修编-.-144cosmxcosnxdx0,m n,m n145sinmxsinnxdx0,m n,m n0,m n1460sinmxsinnxdx0cosmxcosnxdx 2,m n147I2nn0sin xdx20cosnxdxIn 1nnIn2In 1 n 34 2nnn 253（n为大于 1 的正奇数），I11In1 n33 1 nnn2422（n为正偶数），I02-可修编-