常用的積分公式

1、實用標準文案常用積分公式精彩文檔2.(ax b)x=a(1)(ax b)"C(1)3.r(ax + bbln ax+b)+C a（一）含有ax b的積分（a=°）旦 1 Inax+b+C1.' ax + b = a+ b)2 -2b(ax +b) +b2 In ax +b +C4.dx5.x(ax b)-ll n|ax b| Cax b6. dxx2(ax b)=ax b丄弓 ln|ax b| C bx b7.1 (In ax +b + a8.1飛(ax +b 2bln ax +b - a丄)Cax b9. dxx(ax b)2 =b(ax b)lnax b實用標準

2、文案21.）含有-ax b的積分10.、.ax bdx(axCJ= 3a11.2 2 (3ax 一2b) 一 (ax b)3 C12.x2 . ax bdx(15a2x2 -12abx 8b2). (ax b)3 C=105a'13.2(ax -2b). ax b C14.2ax b(3a2x24abx 8b2)、ax b C=15aC (b 0)dx15.2丄 i ax + b “ c、卡列叫盲+C (be。)16.dxJax + bx2 , ax b = bx 2b x . ax bdxax b17."“dx 2. ax b b xdx精彩文檔dxax b22（三）含有x

3、 -a的積分dx1丄 x 小2arcta nCx a = a adxx丄 2n -3 十 dx22、n2 2222、n20 (x +a ) = 2(n 1)a (x + a )2(n - 1)a (x + a )21.2(四)含有ax b(a 0)的積分22.(b 0)(b ： 0)24.2axdxdxax2 b25. dx x(ax2 b)ln2b2xax2 br dx1 adx2 2 212J2dxIn ax +b +C23. ax +b = 2a26. (ax b) = bx b ax br dxaax2 +b 1J 3( 2+x、耳n2+C27. x (ax b) = 2b2x22bx

4、2實用標準文案dx2r dxx+ 丄28'(ax2+b)2 = 2b(ax2+b) 2b'ax2+b2(五)含有ax bx c (a 0)的積分t dxI 229.' ax +bx +c =22ax bc2 arctan .2 C,4ac-b24ac-b22ax b - b2 - 4acInb2 -4ac2ax b b2 - 4ac2(b : 4ac)(b2 4ac)精彩文檔dx2 r戶 dx30.' ax +bx +cIn ax +bx + c 2a ' ax2 +bx + c(六)含有、x2孑(a 0)的積分31.dx.x2 a2 =arsh| C1

5、n(x、) C32.dxx2 a2)3T= +Ca2 x2 a233.i .dx一 x2 a234.x .dx22 3x a )廣x2'a2 C35.36. 2 Cdx - ； x2 a2 -訥(x X2 a2) C2x .dx3x a )-In(xx2 a2) C.x2 a2實用標準文案38.38.39.40.41.42.43.44.dxxix2 a21x2a2 - aIn=adxx2 , x2 a2 =jjx2 +a2dx.(x2 a2)3dxx.x2 a2dx 2 即(XCx(2x2 5a2) .x2=8,(x2a2)3C2 3 4/""22a 8aln(xx

6、a)cJxPx2 +a2dx =評 x2 +a2)Jx2+a2x a dx x2 a2 alnx4ln(x . x2 a2) C82 2X 七a+c-dxI n(x .) Cx= x(七)含有/ _孑(a 0)的積分dxJ r"2245. x -axarch%+GaIn x + Jx2 -a2 +C46.dx22、3x -a )=+ca2 - x2 -a2精彩文檔實用標準文案47.48.49.50.51.52.53.54.55.56.57. 22 dxx -ax *1X2 -a2)3 x= a C-x2L 2x -adx2 ”+ ln x 十 Jx2 _a22x2x(X2 -a2)3

7、xC1aarccos ax精彩文檔j 2jjx2 -a2dx _ fJx2 _a2 號1n x+Jx2 a2JJ(x2 _a2)3 x 8(2x-5a2 ) x2 _a2Cx、x2 -a2dx3J(x2 _a2)3 +c2 2x axdx 4Jx2Jx2 -a2dx _ ；(2x2 -a2)Jx2 a2 亍n x+Jx2-a2 卜 Ca-a arccos x實用標準文案58.X2 -a2+1n x + Jx2 _a2 + C精彩文檔(八)含有、a2 -x2 (a 0)的積分59.t dx* ：=22a -x =.xarcsin Ca60.dx3一 (a2 -x2)61.xdxJ 122a -x

8、 =a2 _ x2 C62.x dxa2 -x2)3 C=.a2 -x263.2 2J f 22 dx-x Ja2 _ x2 + arcsin +Ca -x =22a64.2x .dx223a -x ),2x 2 -arcsin + C a2-x2a65.dxIlnx a2x2 = aa _ a2x2 c66.dxx2 . a2 _ x267.11 a2 -x2dx 2x 22 a . x- a - xarcs inC22a實用標準文案70.2 3a48.x c arcsin C a、.(a2-x2)3dx(5a -2x ) a - x_ 8精彩文檔69.x , a2 -x2dx3'.

9、(aC3a470.Jx2ja2 x2dx _ 才2x2 a2)Ja2 x2 +yarcsi r+C71.a -x dxa2 _ x2a Inx _f2 2aJa x.+c72.22"a -x .2 dxx_73.dx1一 ax2 bx c = . aln 2ax + b +2>/ajax2 +bx +c +C22a -x. xarcsin Cxa(九)含有 -aX bx c (a 0)的積分74.ax2 bx cdx J:a b ®2 bx c2 a ” ax2+ bx + c +C4ac -b2 a3j /一廠x1 Jax2 +bx +c75.ax bx c = a

10、2 a3-In 2ax + b + 2>/aJax2 +bx +c + C實用標準文案77.dx、c bx -ax21. 2ax -bcarcsi n = + C、ab 4ac77. c bx - ax dx2ax - b -、c bx - ax4a2b 4ac . 2ax-b 小 廠 arcs in -C8 一 a3. b2 4ac78.c bx -ax22ax-b c-1 Jc +bx ax2 + 二 arcsin a2 a. b 4ac畀_a（十）含有' x_b 或.(x -a)(b-x)的積分79.(x -b)JxH +(b-a)n(x _b)+C80.xfb xx -a

11、(x -b) .； (b -a)arcsinb xdx81.x _a)(b _x)x - a2arcsi nb_x C (a：b)82.；(x _a)(b _x)dx(a : b)2x -a -b(b -a)2x -a . (x_a)(b_x)arcsin C（十一）含有三角函數(shù)的積分83.sin xdx-cosx Ccos xdxsinx C85.Jtan xdx = -in cosx +C84.精彩文檔實用標準文案86.87.88.89.90.91.92.93.94.95.96.97.Jcotxdx = “ sinx +C精彩文檔secxdxxIn tan（二十一）42+cIn secx

12、+ tanxcscxdxln cscx -cotx +Cxln tan +C2sec xdx=tanx Ccsc2 xdx-cotx Csecx tan xdxsecx Ccscxcotxdx cscx Csin2 xdxx 1si n2x C24cos2 xdxx 1 sin2x C24sinn xdx1.n4n-1. n 2 ,sin xcosxsin xdxnncosn xdx1cosn4xsi nx n 1 cosn xdxnndx _ 1 cosx n _ 2 dxnn -1n-2sin x = n T sin x n T sin x實用標準文案99.sin xn A. cos -

13、xn -1 cosn 2 xdx1n7cos x = n -1dx精彩文檔99.- m n | cos xsin xdxcosmJ xsinn卑 x +-m nm nfm_2- n ,cos xsin xdx1 cosm + xsi nn' x + 1 f cosm xsin m nm nn'xdx100.Jsinax cosbxdx1 1cos(a b)xcos(a -b)x C2(a b)2(a-b)101.sin axsin bxdx1 sin (a b)x 1sin (ab)x C2(a b)2(a -b)102.Jcosaxcosbxdx1 1si n(a b)xsi

14、n (a -b)x C2(a b)2(a-b)103.dxa bsi nx =ata bC/ 22、(a b )104.dxa bsi nx =1.b2 -a2Inata顯 +b _ Jb2 _a22/ 2 | 2(a : b )105.dxa bcosx =丄年 arctan( Jta nx)+C a b 1 a - b- a b(a2 b2)106.dxa bcosx丄 x .a +btan_ +J2 Y-a.x J a + b實用標準文案108.22 取 22 丄 arctan(btanx) Ca cos x b sin x = ababta n x a108.a2cos2x _b2si

15、n2x = 2abbtan x - a109.1 . 1xsinaxdx=尹inaxaxcosax C110.21 222x2 si naxdxx cosax 飛 xs in ax 飛 cosax C=aaa111.11x cosaxdx pcosax - xsin ax C l= aa112.x2 cosaxdxx2sin ax $xcosax -gsinax Cl= aaa（十二）含有反三角函數(shù)的積分（其中 a 0）113.arcsinxarcsinxa =a精彩文檔114.115.x2 arcsin Adxa3xxarcs in =3(x2 2a2) = a2 -x2 Ca 9116.a

16、rccosXdxa實用標準文案118.xarccosdxa2 2X ax()arccos 一24精彩文檔118.x2 arccosxdxa3.x arccos3a 9x 122 -22(x 2a ) . a -x C119.xarcta n -dxax arcta n -ln (a2 x2) Ca 2120.x arctan xdx a2x ax )arcta nx Ca 2121.x2 arctan - dxa3arcta n -3a3a 2 a 22xln(ax ) C6 6（十三）含有指數(shù)函數(shù)的積分122.Jaxdxax CIn a123.eaxdxeax Ca124.xeaxdx-2(

17、ax -d)eax c a125.xneaxdxn axen n -4 ax .x e dx a126.xaxdx實用標準文案128.128.129.130.n x1 n x nx a dxx a -x a dx = lnalnaxnJaxdx1貴 sinbxdx = 胃(asinbx-bcosbx)Ceax cosbxdx12 eax(bs inbx a cosbx) C=a b1eaxsi nn bxdxreaxs in n'bx(asi nbx - nbcosbx)=a +b n2M/sinfxdxa bneax cosn bxdx2 1 2 2 eax cosn d bx(a

18、cosbx nbsin bx)131.l= a +b n2ax n _2e cos bxdxn(n -1)b2斗22a bn（十四）含有對數(shù)函數(shù)的積分精彩文檔132.lnxdx = xln x-x C133.dxxln x =In lnx +C134.xn In xdx(Inx_ 丄)Cn 1135.(In x)ndxx(ln x)nn -1- n (ln x) dx實用標準文案137.xm(ln x)ndx1 mln齊X (X)xm(l nx)ndx精彩文檔（十五）含有雙曲函數(shù)的積分137.shxdx=chx C138.chxdx = Shx C140.141.x 1ch2xdx 二 24Sh2x C（十六）定積分142.cosnxdxsinnxdx 二。139. Hhxdx = |n 乂弋2x 1 ,Jsh x