高等数学常用积分公式查询表

《高等数学常用积分公式查询表》由会员分享，可在线阅读，更多相关《高等数学常用积分公式查询表（17页珍藏版）》请在装配图网上搜索。

1、导数公式:(tgx)sec2 x(ctgx)cscIn2ai 22 一ln(x x a ) C22ja . 22 cln x x a C2 a . x arcsin C 2 a x(secx) secx tgx (cscx)cscx ctgx(ax)axlnaZl 、1(logax) xlna(arcsin x) ,1 x21(arccosx) 2,1 x1(arctgx) 21 x,、1(arcctgx) 21 xdx22sec xdxtgx Ccos xdx2. 2csc xdxctgx Csin x基本积分表：tgxdx In cosx C ctgxdx In sin x C secxd

2、x In secx tgx Cdxcscxdx In cscx ctgx Cdx1x12一 arctg一 Ca xaadx1,lxa w2 lnCx a 2a |x a1 , a xln2a a xsecx tgxdx secx Ccscx ctgxdx cscx Cxx a axdx Cln ashxdx chx Cchxdx shx Cdx.x _arcsin- Cadxln( x,x2a2)C.x2 a222I n sinn xdxcosn xdx0022 , x 22x a dx x a 222x22.x a dx x a2da2 x2dx x Ta2x22三角函数的有理式积分:一.一

3、 2u 1sin x 2, cosx -1 u12幺2， udx -12duu2(一)含有ax b的积分(a0)dxax b a1ln ax2.(ax b) dX1 a(一 (ax1)b)1)3.x-dx ax b1 ,. (ax b ablnaxb)4.2x1ax b1 1/2(axb)22b(axb),2 .b In ax5.dxx(ax b)1ln bax6.dxx2(ax b)7.x .2dx(ax b)8.2-dx9.dx(二)10.12.13.ax b1 a ax b -二lnCbxb2工(In ax a1 , 二 (axa2x(ax b) b(ax b)含有vOxb的积分弋)ax

4、 b2bln ax b4bdx= 7(axb)33ax.ax bdxx2 ax bdx215a22ax bb2axb)(3ax 2b) ,(ax b)3 C105a3(15a2x2 12abx 8b2)、.,(ax b)3 Cxdx = 2 (ax 2bh/axb Cax b 3a14.2一x一.ax b-3 (3ax2 4abx 8b2). ax 15a（四）含有ax b（a 0）的积分15.dxx . ax b16.dxx2 . ax b17.ax b.dx18.而 Vaxb VbL arctan ,bbx2 . ax bax bax b2bC (bC (bdxdxdx0)0)20.dx(

5、x22、n22a )2(n 1)a (x2、n 1a )2n 32(n 1)a2dxT-22xn 1(x a )x % ax（三）含有x2 a2的积分dx1 x19.w2= 一 arctan Cx aa adx2 x i2a22.dx2- ax11 aarctan, x C,ab 1 b、ax 、一 b(b 0)1ln2J ab5/ax bbC (b 0)x aln x a24.2x, x b dx2.dx = -2.axb a a axb25.dx 1 x2= - ln 2x(ax b) 2b ax b26.dx1 a dx x2(ax2 b) bx b ax2 b27.2.dx a , a

6、x b =In3/22 1n2x (ax b) 2b x28.dxx 1 dx(ax2 b)22b(ax2 b) 2b ax2 b2(五)含有ax bx c (a 0)的积分29.dxax bx c2, 2ax b -/2 arctan /2二 C4ac b . 4ac b1b2 4ac2ax b yb24ac ln2ax b 、. b2 4ac(b2 4ac)2C (b2 4ac)30.2xdxax bx cLn ax 2abx cdx2,2a ax bx c(六)含有&a7 (a 0)的积分x.22 一arsh- C1 = ln(x vx a ) C a32.dx x.(x2 a2)3a2

7、 x2 a233 -二 2 dx =& a2C，x a34.XdY=1/ 22X322.(x a )x a35.2、,x2 a2)x . x -# dx = vx,x2 a2236.2 x-27 dx x a )xln(x22 x a.x2 a2)37.dx 1 ,x?xi=a2n、.x2 a2 a C38.dxJx2 a2x2 , x2 a2 a2x39.：=22 , x .22vx a dx = jxa22 aln( x2、x2 a2)40.3422.八a ln( x 、. x a ) C8；2273 .x 22. 7(x a ) dx = -(2 x 5a )vx8xvxa2dx=- 3

8、a2)3 C42.x27xTVdxW(2x2a2) , x2a24ln( x , x2 a2)C843.22、x a-22:-dx = yx ax22,.x a a a In-x44.22x a .2dxx22x a22、ln( x x a )(七)含有Vx2a7 (a 0)的积分45._dx_ xx2=a2 xarch) aC1 = In xJx2 a247.rx； dx = 4x2 a2 C22、x a48.=X dx = -=J=2 C(x上dx 2 3 a2)x a )x a49.=dx= v?a7 a222 aIn250.2ln、x a=-arccosLdx22x x a52.dxv

9、x2 a2x2 , x2 a2a2-53.x2adx=vXa22 a . In256.54.55.33 4 a8In22X 2x vx adx=(2x 8a2),x2a24 a ln 857.a a arccosX58.22x a ,2dxXIn x(八)含有X2 (a 0)的积分22X a ,22dx = 7xaX59.,dx = arcsin C, a2 x2a60.dX(a2x2X3222x ) a . a xx ,/2, dx = V ax22a x62.(a2 x2)dx3_1_a2 x263.2x .dx22a xx .a-x222 a . x arcsin - C64.2x dx

10、22 3a x )x. a2 x2arcsin - Ca65.66.22、a x2a xdx67.a2 x2dx = xvax222a _. x一 arcsin 一68.3a4arcsin-/ 223 x 2 c 2、22(a x ) dx=(5a 2x )vax869.4 3而x2)3 C70.222 , x 2x Va x dx = (2x 8a2) . a2x24a _. x-arcsin 一22- ax. 22dx = ya x x2,a .a xa ln 72.22,a x ,2dxx73.arcsin - Ca含有 a ax2bxc (a 0)的积分 ax2 bx c aln 2a

11、x b 2嘉70bxc22ax b 2一74.Maxbx cdx =vax bx c4a75.,ax2 bx=dx c4ac b2Inbx2ax2axb 2Vavax2bxC C76.dxarcsin2ax b b2 4ac2_；2 , 2ax b 2 b 4ac2ax b 八77.7c bx ax dx =7c bx ax arcsin 、= C4a8, abx axb： 4acx ,1 _；1 b .2 ax b2 dx = 一业 bx ax-= arcsin 2 2Cc bx ax a2 a . b 4ac(十)含有J -xa或J(x a)(b x)的积分a)ln( Jx a Jx b

12、) Ca)arcsinC79. Jxadx = (x b)Ja (b x bx b80. Jxa-dx = (x b)J-a (b ，b x b xdx、(x a)(b x)=2arcsin(a b);-r. 2x a b . (b a) . x a82.V(x a)(b x)dx =(x a)(b x) arcsin J C 44. b x(a b)(十一)含有三角函数的积分83. sin xdxcosx C84. cosxdx = sin x C85.tan xdx = ln cosx C86.cotxdx = In sin x87.secxdxln tan(4x)CIn secx tan

13、x88.cscxdx = ln tan x 2C = lncscxcotx C89.2sec xdx = tan x C90.2csc xdx = cotx Csecx tan xdx = secx C92.cscx cot xdx = cscx93.2 xsin xdx =一 294.2, xcos xdx =一1-sin 2x41 .八一 sin 2x95.2 4n1 n 1sin xdx = sin xcosxnn 2sin xdx96.cosn xdx97.98.dxsinn xdx1cos, n1n 1xsin xcosxn 1n 1 sin x1 sin xnncos x n 1

14、cos99.m ncos xsin xdx100.sin axcosbxdxn 2cos xdxdx_._n 2sin xdxn 2n 1 cos xm 1 n 1 cos x sin xm n1m 1 n 1cos xsin xm n2(a b)cos(a b)x2(am 2 ncos xsin xdxm n 2cos xsin xdxcos(a b)x C b)sin axsin bxdx2(a b)sin(a b)x2(a b)sin(a b)x C102.cosax cosbxdx2(a b)sin(a b)x2(a b)sin(a b)x103.dxa bsin x一 arctan

15、b2a tan2b2)104.dxa bsin xC.a2 b2(a2b2)105.dxa bcosx22 aa babxarctan( -tan )a b，ab2(a2b2)106.dxa bcosx1a ba b. lnb a,xtan2107.dx22T22a cos x b sin x108.dx22,2.2a cos x b sin x109.xsin axdx110.x2 sin axdx1 sin axa1 2111.xcos axdx =112.2 .x cosaxdx（十二）113.,xtan2b aa b(a工 arctan(btanx) abbtan xabtan xaa

16、CC,ln 2ab1xcosax ax cosaxa1- cosax a12、=x sin ax2.2 xsin ax a1 xsin axa23 cosax3 aa含有反三角函数的积分（其中22 xcosax aa 0)2.八 sin ax C aarcsin -dx = x arcsin aa-a2 x2 C22X x a x x -22 c114.xarcsin-dx = ( )arcsin va x Ca 24 a 43d115.2. x x . x 12_ 2. -2x arcsin dx =arcsin (x 2a )7a x a 3 a 9116. arccosdx = xarc

17、cos 4ax2 Caa117.2xarccos dx = (土 a 22a、x x -2 .)arccos a x C4a4122 -22-(x 2a )、a x 93118.2 x , xxx arccos dx =arccosa 3 ax .xa, , 22、一119. arctan dx =x arctan ln( ax ) Caa2x .1,22、，xa-120. xarctan dx=(ax )arctan-xCa2a23 a 22ln( a x ) C632_ x , x _ x a 2 x arctan dx =arctan xa 3 a 6(十三)含有指数函数的积分x .1

18、x 一122. a dx =a Cln a123. eaxdx = 1eax Ca124. xeaxdx=(ax 1)eax Can ax 1 n ax n n 1 ax125. x e dx = x e x e dx a a126.xx x 1 x cxa dx=a 2 a Cln a (ln a)n x1 n x n n 1 x .127. x a dx =x a x a dxln a ln a128. eaxsin bxdx = 2-2eax(asin bx bcosbx) Ca b129. eax cosbxdx = -1-2eax(bsin bx a cosbx) Ca b130.a

19、x . ne sin bxdx1 2 2 eaxsinn 1 bx(asinbx nbcosbx) a bnn(n 1)b2212 2a b nax . n 2 .e sin bxdx（十四）132.133.ax ne cos bxdxn(n 1)b22.2 2a b n含有对数函数的积分ln xdx = xln xdxln ln xax n 1cos bx(a cosbxnbsin bx)ax ne cos2. bxdxx ln x134.xnlnxdx = n(ln135.(ln x)ndx = x(ln x)n六)n 1(ln x)dx136.（十五）xm（ln x）ndx = -1 x

20、 m 1含有双曲函数的积分1(lnx)nxm(ln x)n 1dx137.shxdx = chx C138.chxdx = shx C139.thxdx = lnchx140.（十六）,2 . x sh xdx =2,2 . x ch xdx =2定积分1 .八 sh2x41-sh2x4142.cosnxdxsin nxdx =0143. cosmxsin nxdx = 00, m n144. cosmx cosnxdx =0, m n145. sin mxsin nxdx =,m n146.sin mxsin nxdx0cosmx cos nxdx00, m n147.In = 2 sinn xdx = 2 cos1 xdx n 00InInn 1 n 3n n 2n 1 n 3n n 2（n为大于1的正奇数），I 1=13 1（n为正偶数）,I 0 =一4 2 22x.1 .2 ._23.-2dx =ln ax b Cax b 2a2 22x .a x