常用積分表(絕對有幫助)

1、常用積分公式（一）含有 ax + /?的積分（a工0）dv 1axb a1.= In ax + bC2. j(av + hyidx =3.(ax + /?) + C ( “ h -1 ) 心+1)X|-_dr = (ax + b- bin ax +axrbcr4.1 1dx = (av + h)2 - 2b(ax + /?) +，In lav + h axrba 25=-llnJ x(ax + b) bax + b13.#.dv1a t=1 Inx(ax + b)bx b_7Jdx =-(ln|ax + Z?|4-) + C(av + by cr I 1 ax + hx2J (cix + b)

2、2 adr = (ax + b - 2bn ax + /?| -七)+Cdv9. 1-41nJ x(ax + h) b(ax + h) Irax + h(二)含有JaT帀的積分 2j y/ax + hdx = ylax + b)3 + C 2xJax + bdx =-Ciax - 2b)J(ax + b)3 + CJ15/j x2yjax + bdx =3 (1 Sa2x2 - I2abx + Sb2 jjg + b) + C10.11.12.#.14.A dv =(ax 2b)yjax + b 十 CJ yjax + h 3/#.14.15. Jdvxy/ax + by)ax + b - y

3、/b yjax + h + h2lax + b QlCtSll 4 /+ CE V -b0)vO)16.dvx2y/ax + b_ y/ax + b bT17. j X + & dr = 2yjax + h + bjdrxyjax + b18. Jy/ax + b A yjax + bdv =（三）19.dvxJax + b20.含有x2 f/2的積分dv1x _=arctan + C x +a aadvr av _x2n-3 e dvJ (? + /)2(n-l)a2(x2 +a2),_l + 2(n-l)a2 丿(x2 + a2)/,_l（四）含有ax2 + h（a 0）的積分22.(b0

4、)(bvO)23.324.25.26.27.28.f f dx = - J ar + /? a advdvdvj xax2 + b)dvdvax2 + bxax2 + bdv_fhx baf + b-ln2b2字V+c* 丄 r dr(ar2 + b)2 2b(ax2 + b) 2b ax2 + b(五)含有ax + bx+c (a 0)的積分29fdr22ax + b 小farctan + CJ4ac-b2ax1 + bx + c30.J.1 InJb2 -4ac/4ac-b22ax + b-yjb1 -4ac2ax + b + yjb2 一 4ac(/?2 4ac)；:dx =In ax1

5、 + bx + c af + bx + c 2advf2aax + bx + c(六)含有yjx2 +a2 (a 0)的積分31.=arsli 上 + G = ln( x + yjx1 +a2) + C3432.dv33.535.烏“字 E 號昨+E)+c36.、 J 命嚴(yán)-7?jwE)+c46.47.646.47.#f dv1 vx71 n+ a1 - aJ Xyfx2 + /ina+ C37.38.宀+c39.j y/x2 + a2 dx =字 y/x2 4- a2 + -ln(x + y/x2 -i-a2) + C40.41.jyj(x2 + a2)3dv = (2x2 + Sa2)y/

6、x2 +a2 + -a4 ln(x + yjx2 + a2) + C8 8jxx1 + a2dx = yj(x2 + 6F2)3 4- C42.j+ adv =專(2兀2 + a x，+/ -牛11】(兀+J” + /)+ C43.44.W=_ZZ+吩+c fX46.47.#46.47.#(七)含有y/x2 - a2 (a 0)的積分45.=rch+ =ln 兀 + /x2 - a2 + Cdv46.47.#48.50.49.嚴(yán)E+弓In x + y/x2 - a2 + Cyjx2 -a2 + C748.#48.f dy =larccoS X長Fa52.J宀備+c53.j Jx2 - a2dx

7、 = yjx2 - a1-In x + yjx2 -a1 + C54.-a2 + -a4 In x + y/x2 - a2 + C55.j xyjx1 - a2dx = * J(/ _ /)3 + c56. 4j x2/x2 -a2dx = (2x2 - a2)y/x2 - a2 - -ln x +57.c yjx2 - a2 .ra _Idv = yjx -a -t/arccos+ C58.J fX+ In x + /x2 - a2 + C#48.#48.(A)含有yja2 - x2 (a 0)的積分#48.#48.59.代f噲c60.d#61.861.#61.62.63.x1J啟“送E+和

8、吟+c64.X1產(chǎn)苫吟。65.r av1 . a-yja1 - x/= -InJ x/a2 - x2 adv2+c66.dvx2ja2 -x2I+Ccrx67. _2j yja1 - x2dx =專 J a，_ X +arcsin-+C2a68.69.j（a2 - x2）3dv =畝（/ _ 2x，）Jj _ 牙2 + 討4 arcsin j xja2 - x2dv =_ 疋）3 + c70.用圧 =扣宀心圧7 +詈ag吟+ C71.j也三=需匸7 + dl/一72.（遲竝一逅三a+C J fxa961.#61.（九）含有Ja+ + bx + c （a 0）的積分73| / 山 =-=ln 2

9、cvc 十 b 卜 2 xfa yjax1 + bx+ c + C J yjax2 + bx + cJa#74.yax2 + bx + cdx =加 十J ax + bx + c4ac - bJ4aIn 2ax + b + 2y/aJax2 + bx + c + C75.f dx =丄 yjax2 + bx + cJ yjax2 + bx + c a- In 2ax + /? + 2aJax2 +bx- c + C 2如76.dvJc + bx 一 ax1. 2ax-b _=arcsin j= + CJa yjb2 + 4ac77.j yjc +bx-ax2dx =彳I:r h1 + 4ac

10、. lax-h yjc + bx-ax +arcsin - z+ C4。8 J/y/b2 + 4ac78.rx.1 I:r b. 2ax-b 小/dr =jc + bx-ax + arcsin z+ CJ yjc + bx- ax2a2f Jb +4ac（十）含有J士呂 或Ja-a）（b- X）的積分79.三智= (x-+ (b-a) ln(7x-a + yjx-h) + C80.產(chǎn) = (x -+ (b _ a) arcsinb-x b-xb-xb-x81.dx_/3= 2arcsin*/ + C (a b2)J a + hsin x yj-b1y/a2 - b1dv1 Ina + b si

11、n x yjb2 - a2a tan + /?- yjb1 - a22a tan 4- b + yjb1 - a1+ C (/ b2)a + b .Ina + b cos x a + b7 b-advx a + b tail f J 2 b-a a + b b-a+ C (a2 0)113. arcsin= x arcsin + yja2 - x2 + CJaadva + barctan(abcosx a + ba-bab114.f a arcsindv = ( 一 ) arcsin+ yjcr - x2 + C Ja24a 4115.a2 arcsindv = arcsin + (x2 +2

12、ci)Ja -x2 +C Ja3a 9116.arccosHiv = xarccos- lcr - x2 + CJaa117.f Aarccosdv =) arccos- x2 + C1a24a 4118.f x2 arccos dv = arccos - (x2 十 2a)y/a2 - x2 + C Ja3a 9119.arctandv = x arctan- ln(a2 + x2) + CJaa 2120.fXt1,22、Xaj1 x arctan dv = (/ +arctanx+ CJa2a2121.f x2 arctan dv = arctan - x2 + ln(a2 + x2)

13、+ C Ja3a 66（十三）含有指數(shù)函數(shù)的積分122.t/dr = -aA + CJIn ci123.f cadv =丄嚴(yán) + CJa124.f Ae/Vdr =丄(ax - l)ea + CJcr125.Aweavdv =丄 xW - xn-leaxdxJaaJ126.f Aadr = ax - a + CJlutz(Ina)2127.f xnaxdx - 1 曲 n xn-axdxJIn aIn a128.j en sin bxdx = 歹 e (a sin bx-h cos bx) + C129.j en cos bxdx = e (b sin bx + a cos bx) + C15

14、130eav sinn bxdx = ea sin721 bx(a sinbx 一 nhcosbx)Jcr + bx+(嚴(yán)si】嚴(yán)加ckcr + br J131.J eM cos hxdx =1a2 + b2n2 Ccos7/?A：(d coshx + nb sinbx)143.16143.#+呷-”叮嚴(yán)cos%皿 a + b-n(十四)含有對數(shù)函數(shù)的積分132. J In xdx = xlnx- x + C133. 山=In lnx + CJ xlnx134 xn In xdv = - x,+l (In x) + CJh+1n+1135. j(In x)ndr = x(ln x)H - n j(Inx/dx136. f xm(nx)ndx = x/M+l(lnx)/, - f /M(lnx)n-ldvJm + 1zn +1(十五)含有雙曲函數(shù)的積分137. jsludv = cliv + C138. J clivdv = sliv I C139. Jtludv = Incliv + C140. I sh2xdr = 一一+ sh2x + CJ24141. ch2xdv = + sh2.r + CJ24(十六)定積分142. f cosnxdx = f sinn.vcLv =0J