關(guān)于高等數(shù)學(xué)常用積分公式查詢表

1、導(dǎo)數(shù)公式:(tgx) (ctgx) (secx) (cscx) (ax)(lOga X)2 sec x2csc xsecx tgxcscx ctgxaxln a1)xlnatgxdx In cosx Cctgxdx Insinx C secxdx In secx tgx C cscxdx In cscx ctgx Cdx2 2 a xdx22x a-arctg - Ca1 In 2adx 11axe2 In Ca x 2a a x dx；22a xarcsin- Ca(arcsin x)(arccosx)(arctgx)(arcctgx)22In sinn xdxcosn xdx00Jx2 a

2、2dx x Vxa2/ 22 , x 22x a dx - x a 222 x 22a x dx 一 ： a x 2基本積分表：三角函數(shù)的有理式積分：（一）含有ax b的積分（a 0）dx2- cos xdx.2 sin x2sec xdx tgx C2csc xdx ctgx Csecx tgxdx secx Ccscx ctgxdx cscx Cxx aaxdx C In ashxdx chx Cchxdx shx Cdx -22、八22 ln(x x a ) C. x aIn231mx x2 a2)2,a I 廣 201nx *x a C22 a . x 一arcsin - C2 a2.

3、2.3.4.5.6.7.8.9.二）10.12.13.14.15.dxax(ax-=1ln ax b C b a1 b) dx =a( 1)xax bdx=2(ax ba2x-dxax bdxx(ax b)dxx2(ax b)(axblnb) 1 C (ax b) C1)1 1 /-2(axb)2_22b(ax b) b Inax bU7dx2x(ax b)2dxdx1ln b1bxaxax b1 g(axaax：) ax b2b In ax bb2ax b2x(ax b) b( ax b)-12 ln bax bTax_bdx = Jax b)33ax、ax bdxx2 . ax bdx21

4、5a22(3ax 2b) . (ax b)3 C105a3(15a2x2 12abx 8b2) .(ax b)3dx=dx bdxx > ax b2 ,一2 (ax 2b),ax b C 3a2r (3a2x2 4abx 8b2) ax 15a- ax b , bVbVaxb 瓜C (b0)arctanax bbC (b0)17.dxx ax bbx 2b x , ax bax b. 八n , dx17.dx = 2jax b b /xx %, ax b“ c. axb , axb a dx18. 2dx = xx 2x < axb（三）含有x2 a2的積分dx 1, x19.-2

5、 = 一 arctan-x a a a20.dx _x, 22 . n 2 22 . n 1(x a )2(n 1)a (x a )2n 3""22(n 1)adxr22-ni(x a )d = "x a Cx a 2a |x a2（四）含有ax b（a 0）的積分22.dx2Tax b271n(b 0)C (b 0)23.xdx = - ln ax2 b C ax2 b 2a24.2x, x b dx_rdx = _ _2 人axb a a ax b25.dx 1 , x2= - ln -2x(ax b) 2b ax b26.dx1a dxoZ = - Zx (

6、axb) bx b axb27.2.dx a , ax b2 rr - o. 2ln 2x (ax b) 2b x12bx228.dxx 1 dx= 22_2_2(ax b) 2b(ax b) 2b ax b.2（五）含有axbx c (a 0)的積分29.dxax bx c2、,4ac b1， 2ax b - a arctan- C2.4ac b2(b2 4ac)30.x 72dxax bx c(六)含有x2a (a.b2 4ac12aIn ax20）的積分In2ax b b2 4ac2ax b b2 4acbx cb dx22a ax31 - dx = arsh- C1=ln(x V?)

7、C ,x a a32.dxx = -(x2 a2)3a2 , x2 a234.35.36.37.38.bx c(b2 4ac)33.C1x3dx.(x2 a2)22a, x2 a2dx =2a /ln( x2.x2a2)223x a )dxx x2 a2dxdx, x2 a2ln( x11n a.x2 a2 aC 239-txadx=Vx2a-2 -in( x &2a_2) c2240.41 .42.5/(xa2)3dx = -x (2x2 5a2) Vxa xVx2a2dx = V(x2a2)3 C3a41n(x . x2 a2) C 8/收 a2dx 臂(2x24a221n( x

8、. x a ) C844.XC-2222“x ax a22、o44.-dx = ln(x vx a ) CXx含有 x2 a2(a 0)的積分45.dx22x aarch * Ci=lna46.dx.(x2 a2)3x222a x a47.j 2x 2 dx = Jx2 a2 C .x a49.50.dx48.2 x22x a x2 a22-ln2、x2 a2.(x2=dx 2 3a )ln x x2 a2:dx= 1arccos 1a2a xx. x2 a2 a |x52.dx2 -22x 、x a , 2,53- Vx2adx = x2 a2 ln x xx C2254- 7(x2a2)x

9、 a ,-22a -dx = x(2x2 5a2)Vx2a2 3adx = V x a a arccos 口 Cxx1nx Vx2a2 C-8855- xVx2a2dx = x/(x2a2)3 C456- x24xadx = (2x2 a2)Vx2a2 ln x xya2 C57.8859.(八)59.60.62.63.64.65.66.67.69.70.72.22x a .2dxxdx22,a xdx2：x (a 0)的積分arcsin - Ca.(a2 x2)3In x2 axa2 x2dx.(a2 x2)2x22,a xdx x a2dxx2、a23dxdx23x )dx,a2x.a72

10、x,a21lna aa 一 x arcsin 一arcsin - Ca、a2x222a x2a x(a2 x2)3dx = x (5a2x7a2x2dx = 工 V(02368.22x . a2 . x _ 2x dx = - (2x8, a x魂 2dx = vaxx一22 a x ,2dxxa 一一一 x一 arcsin -2x2) - a2x2)3 Ca2) a2aln_4 . x a arcsin-aa . x 一arcsin - C' 22-aX- Carcsin - Ca(九)含有 a ax2bxC (a 0)的積分73.dx.ax bx cIn 2ax b 2 Va Va

11、xbxc C74.ax2 bx cdx2ax b 2-, ax bx c4a75 . j 2 x dx = 1 Vaxbxc ax bx c a76.dx. c bx ax1. 2ax barcsin 2=、a b 4ac78.、c bx ax2dx2ax b 2c bx ax4ab2 4ac .arcsin8 a32ax b4aci x 2dx,c bx ax2/cbx_ax2 ab . 2ax b-=arcsin:2 a3 b2 4ac(十)含有 J Ja或J(x a)(b x)的積分79.x a ./dx = (xx bb)'.x b)ln(而a|尿b|)80.a)arcsind

12、x.(x a)(b x)2arcsin x-a b xC (a b)82./(xa)(bx)dx =2x a bj(x a)(b x)Zarcsin4(十一)含有三角函數(shù)的積分83. sin xdx = cosx C84. cos xdx = sinx C85. tan xdx = In cosx C86. cot xdx = In sinx Csecxdx = In tan(-4x、-)C2In secx tanx一. x _.-88. cscxdx= ln tan C = In cscx cotx C2一289. sec xdx = tanx C290 csc xdx = cotx C91

13、. secx tan xdx = secx C92. cscx cot xdx = cscx C93.94.95.96.97.98.99.2.sin xdx2.cos xdxx 1 . sin2x2 4x 1 . 0 sin2xnsinxdx =1n 1_ _-sin xcosx nnn1n 2一 sin xdxcosnxdx = nn 1 cos xsin xn1 nn 2cos xdxdx1 cosxn2dxn sinx nn11 sin xn1:一 n 2sin xdx=1sin xn2dxncosx nn 11 cos xn1n 2cos x2 4c“m n1m 1 n 1 m 1co

14、s xsin xdx =cos xsin x m nm n1m 1 n 1 n 1=cos xsin x m nm nm 2. ncos xsin xdxm n 2cos xsin xdx100.sinax cosbxdx =cos(a b)x2(a b)12(a b)cos(a b)x Csinax sinbxdx1 sin(a 2(a b) 'b)x1 sin(a b)x 2(a b) ')102.cosaxcosbxdx1sin(a b)x2(a b)一1一sin(a b)x 2(a b),x .a tan- bdx222103-=r ) arctan J、 C (a b

15、 )a bsinx .a2 b2,a2 b2104.dx1a bsin x , b2 a2,x22a tan b b a2a tan b “b2 a2 22 2-2、(a b ) dx2 ab * / a_b , x、八 22105- =Jarctan(Jtan-) C (a b )a bcosx a b a ba b 2106.dx 1 =a bcosx a bInC(a2b2)107.dx1, ,b,、-222-= arctan( tanx)a cos x b sin x ab a108.dx1 , b tan x a222=lna cos x b sin x 2ab btan x a10

16、9.110.111 .xsinaxdx = sinax - xcosax Caax2 sin axdx1 2 -x cosax a22 xsin ax a: cosax C3 axcosaxdx = cosax -xsinax Caa2.1 2_._2 2112. . x cosaxdx = - x sin ax - x cosax sin ax C aaa(十二)含有反三角函數(shù)的積分(其中 a 0)113. arcsimxdx= xarcsin Va2” Caa22x ./ x a、. x x -2 八114. xarcsin dx =( )arcsin va x Ca 24 a 4115.

17、32 x, _ xx 1 /、,2x arcsin -dxarcsin - - (xa 3 a 9C116. arccosxdx = xarccos- 4ax2 C aa22x,xa 、 xx 22c117. x arccos-dx=()arccos vaxCa24 a4118.2 x x xx arccos dx = 一arccos-a 3 a9(x2 2a2), a2 x2119.120.一一一x.x a. , 22、arctandx = xarctan- -ln(a x ) Caa 2,x .1/2x arctan- dx = - (a a 2x2)arctan- ax Ca 232.

18、x , x x a 2x arctan dx = arctan xa 3 a 63a 2ln( a6x2) C（十三）含有指數(shù)函數(shù)的積分V .1 Y 一122. axdx=ax CIna1 123.eaxdx = 1 eax Ca1124. xeaxdx = (ax 1)e Can _ ax1 n ax n n 1 ax .125. x e dx = - x e - x e dx a ax x x 1 x126. xa dx =a 2 a CIn a (In a)n x1 n x n n 1 x ,127. x a dx =x a x a dxIn a In aax .1 ax 一一 一 一一 一128. e sin bxdx = 2-e (asinbx bcosbx) Ca b1129. e cosbxdx = 2e (bsinbx acosbx) Ca b1130. e sin bxdx = -2-e sin bx(asinbx nbcosbx)a b nax n1 ax n 1 一 一 一 一 一 一 一131. - e cos bxdx = 2-e cos bx(acosbx nbsinbx)a b n(十四)含有對數(shù)函數(shù)的積分132. ln x