高数积分公式大全

1、高数积分公式大全常用积分公式()含有6 +的积分("0)1.1心 =ax+ bIn |r/x + Z?| + C2.| ax + b)"dx(+ 1)(ax + b)1( WT )3.j -ydr(ax + b-bnix + /?|) + C4.fd.vJ ax+ b(a" + b)? - 2b(ax + b) + b2 In |t/x + | + C5. =-lln| + CJ x(ax + b) b x 6. 竺斗CJ x(cix + b)bx x I7J/冲=5。皿*+七)+。8 -7dx-(a + b-2bnax + b )+ Cj 3 + )2/11 a

2、x + b9 r dr 11 f cix + b.r -r In + CJ x(ax + by bax + b) lr x I(二)含有G的积分11* J x4ax + bdx 10 | y/ax + bdxQ(ax + b)' + C二. (3ax-2/?)J(ax + b)' +C12 jx2 Jax + bdx2、H； l(15a2八二-I2abx + 8l/)yj(ax + b)3 +C13. f "J /ax + b2<Lv 二r(ax - 2b)yax + b + C 3a -14. f 1dx =(3t/2x2 - 4abx + Sb2 )Jax

3、+ b + CJ y/ax + b 15"、15-cLvxyjax + b3nl 尸一"| + C fb yjcix + b + y|2lax + b -Q 7 -b3>o)S < o)16.17.dxJax + ba r dxx2y/ax + bbx 2b)x&tx + bJax + b ,_dvx2cix + b + “ J x-Jax + bg : >/>x + /?d( yjax + b + a f 心J x2x 2 J xy/ax + b(三)含有的积分19.f ax _ 1x-7arctan + CJjT+a. aa20.dr _

4、x(Y +/) 一 2(_1)。2*2 +/产2n -3 r dx+ 2(n-l)«2 J (x2+r/2),M21. f = ±lnM + CJ X* -cr 2a x + a（四）含有ax2 +b(a> 0)的积分22.i= arctan Jx + C 牺 bIn>ax-yb y/ax + y/bS>o)(b < 0)23.f dx- In ax2 +b +C'+ h2a24.dv25.26.27.28.x h dx -、a aax + bdrx(ax2 + b)=扑xax2 + b+c dxJ x2(ax2 +b)r dxJ x(ix2

5、 +b)1 a r dvbx hax2 + b-rln 2lrax1 + bA+cdv(ax2 + b)22bM2 + b) + 2bax2 + b（五）含有a/+/zr + c （。）的积分lax + b29. Jdx( z arctan f z y4ac - b2- b2(h2 < 4ac)ax2 + bx + clax + b - J 护- 4ac/ , 三 Inyjb2 -4ac 2cix + b + lb2 -4ac+ C (b2 > 4ac)30. Jax1 +bx + cdx In ax2 +bx + c2ai-±f , 2a J。厂 + bx + c（六）

6、含有G7-。）的积分dv=arsh + G = n(x + lx2 +a2) + C a32. = J *J d(x2 +c)3a2/x2 +a233. .f . A . dy jx2 +a +C34. f -=L=dv = -/1，+ cJ J* +1)3 G +/35 j j ： dx3 yjx2 +a2 - * ln(x + /x2 +a2) + C36 J /，、=dx = -= + ln(x + 5/x2 +a2) + C y(X2 +/)3<X2 +C1240 . f J(x2 +M)& = (2,v2 + 5a2)jx2 +a2 +-a4 ln(x + Jx? +/

7、) + C J8841 . Jxx，+4cU =: -+，)' + C342 jx2yjx2 +a2dx (2x2 + a2)J/+/ -ln(x + ,x2 +()+ Cr-Jx2 +a2 . Lr , vx +a2 - a dv W +U +an-： + CJ xkl44. f'1" dv = -*'+ ln(x +4x2 +/) + CJ X"X（七）含有/?=3。）的积分53.Jyfx2 -c/2cLv = /x2 -a2 - - In x + 22>/x2 -a2 +C54.-/)& = (2x2 - 5ci2)/x2 - a

8、2 +-aA In x + x88-a2 +C55.|dr =: g y(x2+ C56.| x2jx2 -«2(iv； (2x - a2)5/a-2 - a1 -ln x + 4x2 -Cl2 + Cr a cr -aarccos + CxSQ rx2-a2 . VP-a2 .ro>dx + ln x + yx -cr +CJ 尸x（八）含有k7m。）的积分61 1,，= -7«2-x2 +c 'yja2 -x267.64.65.66.y/a2 -x2dx+ arcsin + CJ22 a68. V(2 -x2fdx = -(5a2 - 2x2)yja2 -

9、x2 +-«4 arcsin - + C J88 a69.Jxy/a2 -x2dx-储一百+c70.J x2-x2dr =: "(2x2 - a2 )la2 -x + arcsin + C71.i L r . (i Ja 一厂 dx J(r 一尸 + a In+ C72.业上Mexa闵(九)含有Gax-bx + c ("0)的积分73.dxyjax2 +bx + c5= In lax + b + 2fa cix1 +bx + c + C74.j vox1".lax + b+ bx + cdx -yjcix2 +bx + c+ 4aIn 12ax + 0+

10、2ay/ax2 +bx + c + C75.xyjax2 +bx + cdr fax2 +bx + ca；(： In 120r + b + 2Ja jax2 +bx + c + C76.dxyjc + bx-ax21. 2ax-b 厂=arcsin / 、+ CG lb2 +4ac77 | c + bx-ax2dx2ax - b rtr lr + 4ac . lax 一 一yjc + bx- axr d-=r- arcsin )+ C4"8，/ yjb2+4ac78.Jc + bx - ax?.1 I；T b . 2ax-bdr Jc + bx - ax += arcsin ,- +

11、 Ca2y/(J Jb2 +4ac(+ )含有低或J()S 7)的积分79. J序=(i80. J层层+ (Z?-6/)arcsin后I+c81.dx(x-a)(b-x)2arcsin层+c(a < b)82 j- a)(b - x)dx83.84.85.86.87.j sec xdx = In/兀 x、 tan(+ )4 2+ C ln|secx + tan x| + C88.f esc xdx In tan J2+ C In |cscx-cot x| + C2'：-" yj(x-a)(b-x) +(b:)arcsin+ C(a < b)(十一)含有三角函数的积

12、分jsinxdx -cosx + Cj cos xdx sin x + Cj tan xdx = 一 In |cos x + C f cot xdx = ln|sin x + C89.90.j sec2 xdx - tan x + Cfesc2 xdx "cot x + C91.J sec x tan xdx sec x + C92.JcscxcotxcLv -cscx + C93.sin2 xdxsin2x + C2 494.7.X 1 3 一cos" xax + sin2x + C2 495.n .1 1 1 f .sin xdx - - - sin x cos x +

13、 sin - xaxnn J96. *1-1 1 r cos xax cos xsinx + cos xaxnn J97.r dxJ sin" xcos x n - 2.“I + sin xdv-1 J sin”" x98.d.vCOS Xsinx -2 r- +n - 1 cos" x - 1 J cosdv99.J cos，" xsin' xdx "!cos”“ xsin,*l x + - f cos"“ xsin" xdx m + nm + n J7+l “一I " - 1 C m “-2cos xs

14、in x + cos xsin xdx m + nm + n J100.j sinaxcosbxdxcos(a + b)xcos(a-b)x + C 2(a + b)2( 一 b)101.sin cix sin bxdx -!J2(a + b)sin(6/ 4- b)x + -!y-sin(tz - b)x + C102.jcos6/xcosZ?xcU !sin(6/ + b)x +!sin(a - h)x + C 2(a + )2(d)103.X foa tan +。arc tan .二 三F C(1> h")77104.105.dxa + bsnxxdv2+ C (a2 &

15、lt;b2)Ina + h sin x2f + C (a2>h2)J a+ bcosx106.d.va + bcosx1 a + b x Jn a+bb-ax la + b 血5 +Jx la + b tan2-+ C (a2 <b2)107._、心，J a2 cos x + b2 sin*_ = ±arctan(-tanx) + C x ab a108.dxJ «2 cos2 x-b2 sin2ilnb tan x + ab tan x-a+c109.卜 sinc/xdx -'Lsinax-Lcosax + C cr a110.|x2sinvd

16、5; 一-x2 cosax+ xsincix + cosax + C acrcr111.J xcoscixdx -coscix + -xsinax + C 武a112.Ycoscmk = -x2 sint/x + 4xcos - Asin64X + Caa（十二）含有反三角函数的积分（其中。）arcsinx arcsin +- x2 + C114.|xarcsinJa- (-)arcsin- + -j+ C115.f x2 arcsin iiv = arcsin- + -(x2 + 2a2)yja2 -x2 + CJa 3 a 9116.f arccos Aij a arccosJ a117.

17、f x arccos -dvJMa，x x r -(一一一)arccos- >/«* 一厂 +C 24a 4118.， x . x厂 arccos dx arccos-(x2 + 2a2)- a2 + C a 9119.x f X 41/) 八 一arctan d.v xarctan - - ln(r7" +x") + C aa 2120.X j1)八X Clcx arctan dx (cr +x) arctanx + Ca2a 2121.>X .XX Cl i i / 1，、一厂 arctan dr arctanx din(cr +x)+ Ca 3

18、a 66（十三）含有指数函数的积分122. J& = jcJe，dA = leax+c124.- -4-(t/x- l)ea' +CJcr125.ax 一-a e axa J126.ax-ax+CIna (lna)一127.xnaxdx! xnanaIn axnaxdx128.|enr sin bxdx 7 c°xa sin bx-b cos bx) + C cr +Zr129.J e" cos bxdx - e“" (Z? sin bx + a cos bx) + C cT +130.Jearsin, bxdx .丁二产 sin”' hx(a sin bx - nbcos bx)n(n -1)/72+ a2 +b2n2e"v sin" bxdx131.丁r e"v cos"】 bx(a cos bx + nb sin bx)e" cos" bx(x ", cr+lrnn(n -1)/?2?737-CL+)-J