常用函數(shù)積分表(增強(qiáng)版)

1、1. fx+gxdx=fxdx+gxdx2. fx-gxdx=fxdx-gxdx3. fxdgx=fxgx-gxdfx4. axdx=axlna+C，a1，a>05. xndx=xn+1n+1+C，n-16. 1xdx=lnx+C7. exdx=ex+C8. sinxdx=-cosx+C9. cosxdx=sinx+C10. sec2xdx=tanx+C11. csc2xdx=-cotx+C12. secxtanxdx=secx+C13. cscxcotxdx=-cscx+C14. ax+bndx=ax+bn+1an+1+C，a0，n-115. dxax+b=1alnax+b+C，a01

2、6. xax+bndx=ax+bn+1a2ax+bn+2-bn+1+C，a0，n-1,-217. xax+bdx=xa-ba2lnax+b+C，a018. xax+b2dx=1a2lnax+b+bax+b+C，a019. x2ax+bdx=12a3ax+b2-4bax+b+2b2lnax+b+C20. x2ax+b2dx=1a3ax+b-2blnax+b-b2ax+b+C21. x2ax+b3dx=1a3lnax+b+2bax+b-b22ax+b2+C22. x2ax+bndx=1a3-1n-3ax+bn-3+2bn-2ax+bn-2-b2n-1ax+bn-1+C，n1,2,323. dxxa

3、x+b=1blnxax+b+C，b024. dxx2ax+b=-1bx+ab2lnax+bx+C25. dxx2ax+b2=-a1b2ax+b+1ab2x-2b3lnax+bx+C26. xax+bdx=215a23ax-2bax+b32+C27. x2ax+bdx=2105a315a2x2-12abx+8b2ax+b32+C28. ax+bndx=2ax+bn+2an+2+C，a0，n-229. xnax+bdx=2a2n+3xnax+b32-2nba2n+3xn-1ax+bdx 循環(huán)計(jì)算30. ax+bxdx=2ax+b+bdxxax+b=2ax+b-2barctanhax+bb+C31.

4、 dxxax+b=2-barctanax+b-b+C，b<032. dxxax+b=1blnax+b-bax+b+b+C，b>033. ax+bx2dx=-ax+bx+a2dxxax+b+C34. ax+bxndx=-ax+b32bn-1xn-1-2n-5a2bn-1ax+bxn-1dx，n1 循環(huán)計(jì)算35. xnax+bdx=2a2n+1xnax+b-bnxn-1ax+b+C 循環(huán)計(jì)算36. dxx2ax+b=-ax+bbx-a2bdxxax+b+C，b037. dxxnax+b=-ax+bbn-1xn-1-2n-3a2bn-1ax+bxn-1dx，n1 循環(huán)計(jì)算38. xnax

5、+bdx=22n+1xn+1ax+b+bxnax+b-nbxn-1ax+bdx+C 循環(huán)計(jì)算39. dxa2+x2=1aarctanxa+C，a040. dxa2+x22=x2a2a2+x2+12a3arctanxa+C，a041. dxa2-x2=12alna+xa-x+C=1aarctanhxa+C，a0，a>x42. dxa2-x22=x2a2a2-x2+14a3lnx+ax-a+C43. 1x2-a2dx=12alnx-ax+a+C=-1aarccothxa+C，a0，x>a44. dxa2+x2=lnx+a2+x2+C45. a2+x2dx=x2a2+x2+a22lnx+

6、a2+x2+C46. a2+x23dx=xa2+x234+38a2xa2+x2+38a4lnx+a2+x2+C47. a2+x25dx=xa2+x256+524a2xa2+x23+516a4xa2+x2+516a6lnx+a2+x2+C48. xa2+x22n+1dx=a2+x22n+32n+3+C49. x2a2+x2dx=x8a2+2x2a2+x2-a48lnx+a2+x2+C50. x2a2+x23dx=xa2+x256-a2xa2+x224-a4xa2+x216-a616lnx+a2+x2+C51. x3a2+x2dx=a2+x255-a2a2+x233+C52. x3a2+x23dx

7、=a2+x277-a2a2+x255+C53. x3a2+x22n+1dx=a2+x22n+52n+5-a2a2+x22n+32n+3+C54. x4a2+x2dx=x3a2+x236-a2xa2+x238+a4xa2+x216+a616lnx+a2+x2+C55. x4a2+x23dx=x3a2+x258-a2xa2+x2516+a4xa2+x2364+3a6xa2+x2128+3a8128lnx+a2+x2+C56. x5a2+x2dx=a2+x277-2a2a2+x255+a4a2+x233+C57. x5a2+x23dx=a2+x299-2a2a2+x277+a4a2+x255+C58

8、. x5a2+x22n+1dx=a2+x22n+72n+7-2a2a2+x22n+52n+5+a4a2+x22n+32n+3+C59. a2+x2xdx=a2+x2-alna+a2+x2x+C=a2+x2-aarcsinhax+C60. a2+x23xdx=a2+x233+a2a2+x2-a3lna+a2+x2x+C61. a2+x25xdx=a2+x255+a2a2+x233+a4a2+x2-a5lna+a2+x2x+C62. a2+x277dx=a2+x277+a2a2+x255+a4a2+x233+a6a2+x2-a7lna+a2+x2x+C63. a2+x2x2dx=lnx+a2+x2

9、-a2+x2x+C64. x2a2+x2dx=-a22lnx+a2+x2+xa2+x22+C=-a22arcsinhxa+xa2+x22+C65. dxxa2+x2=-1alna+a2+x2x+C=-1aarcsinhax+C66. dxx2a2+x2=-a2+x2a2x+C，a067. dxa2-x2=arcsinxa+C，a0，xa68. a2-x2dx=x2a2-x2+a22arcsinxa+C，a0，xa69. a2-x2dx=12xa2-x2-sgnxarccoshxa+C，xa70. xa2-x2dx=-a2-x233+C，xa71. x2a2-x2dx=a48arcsinxa-1

10、8xa2-x2a2-2x2+C，a072. a2-x2xdx=a2-x2-alna+a2-x2x+C，xa73. a2-x2x2dx=-arcsinxa-a2-x2x+C，a074. x2a2-x2dx=a22arcsinxa-xa2-x22+C，a0，xa2-x275. dxxa2-x2=-1alna+a2-x2x+C，a076. dxx2a2-x2=-a2-x2a2x+C，a077. dxx2-a2=lnx+x2-a2+C78. x2-a2dx=x2x2-a2-a22lnx+x2-a2+C79. x2-a2ndx=xx2-a2nn+1-na2n+1x2-a2n-2dx，n-1 循環(huán)計(jì)算80

11、. dxx2-a2n=xx2-a22-n2-na2+n-32-na2dxx2-a2n-2，n2 循環(huán)計(jì)算81. xx2-a2ndx=x2-a2n+2n+2+C，n-282. x2x2-a2dx=x82x2-a2x2-a2-a48lnx+x2-a2+C83. x2-a2xdx=x2-a2-aarcsecxa+C=x2-a2-aarccosax+C，a084. x dxx2-a2=x2-a2+C85. x dxx2-a23=-1x2-a2+C86. x dxx2-a25=-13x2-a23+C87. x dxx2-a27=-15x2-a25+C88. x dxx2-a22n+1=-12n-1x2-

12、a22n-1+C89. x2-a2x2dx=lnx+x2-a2-x2-a2x+C90. x2x2-a2dx=a22lnx+x2-a2+x2x2-a2+C91. x2x2-a23dx=-xx2-a2+lnx+x2-a2a+C92. x4x2-a2dx=x3x2-a24+38a2xx2-a2+38a4lnx+x2-a2a+C93. x4x2-a23dx=xx2-a22-a2xx2-a2+32a2lnx+x2-a2a+C94. x4x2-a25dx=-xx2-a2-x33x2-a23+lnx+x2-a2a+C95. x2m dxx2-a22n+1=-x2m-12n-1x2-a22n-1+2m-12n

13、-1x2m-2x2-a22n-1dx+C=-1n-ma2n-mi=0n-m-112m+i+1n-m-1ix2m+i+1x2-a22m+i+1，n>m096. dxx2-a23=-xa2x2-a2+C97. dxx2-a25=1a4xx2-a2-x33x2-a23+C98. dxx2-a27=-1a6xx2-a2-2x33x2-a23+x55x2-a25+C99. dxx2-a29=1a8xx2-a2-2x33x2-a23+3x55x2-a25-x77x2-a27+C100. x2x2-a25dx=-x33a2x2-a23+C101. x2x2-a27dx=1a4x33x2-a23-x55

14、x2-a25+C102. x2x2-a29dx=-1a6x33x2-a23-2x55x2-a25+x77x2-a27+C103. dxxx2-a2=1aarcsecxa+C，a0104. dxx2x2-a2=x2-a2a2x+C，a0105. dxax2+bx+c=24ac-b2arctan2ax+b4ac-b2，4ac-b2>0106. dxax2+bx+c=2b2-4acarctanh2ax+bb2-4ac=1b2-4acln2ax+b-b2-4ac2ax+b+b2-4ac，4ac-b2<0107. dxax2+bx+c=-22ax+b，4ac-b2=0108. dxax2+b

15、x+c=12alnax2+bx+c-b2adxax2+bx+c+C109. mx+nax2+bx+cdx=m2alnax2+bx+c+2an-bma4ac-b2arctan2ax+b4ac-b2+C，4ac-b2>0110. mx+nax2+bx+cdx=m2alnax2+bx+c+2an-bmab2-4acarctanh2ax+bb2-4ac+C，4ac-b2<0111. mx+nax2+bx+cdx=m2alnax2+bx+c-2an-bma2ax+b+C，4ac-b2=0112. dxax2+bx+cn=2ax+bn-14ac-b2ax2+bx+cn-1+2n-32an-14

16、ac-b2dxax2+bx+cn-1+C113. xax2+bx+cndx=bx+2cn-14ac-b2ax2+bx+cn-1-b2n-3n-14ac-b2dxax2+bx+cn-1+C114. dxxax2+bx+c=12clnx2ax2+bx+c-b2cdxax2+bx+c+C115. dxax2+bx+c=1aln2a2x2+abx+ac+2ax+b+C，a>0116. dxax2+bx+c=1aarcsinh2ax+b4ac-b2+C，a>0，4ac-b2>0117. dxax2+bx+c=1aln2ax+b+C，a>0，4ac-b2=0118. dxax2+b

17、x+c=-1-aarcsin2ax+bb2-4ac+C，a<0，4ac-b2<0119. dxax2+bx+c3=4ax+2b4ac-b2ax2+bx+c+C120. dxax2+bx+c5=4ax+2b34ac-b2ax2+bx+c1ax2+bx+c+8a4ac-b2+C121. dxax2+bx+c2n+1=4ax+2b2n-14ac-b2ax2+bx+c2n-1+8an-12n-14ac-b2dxax2+bx+c2n-1+C 循環(huán)計(jì)算122. x dxax2+bx+c=ax2+bx+ca-b2adxax2+bx+c+C123. x dxax2+bx+c3=-2bx+4c4ac

18、-b2ax2+bx+c+C124. x dxax2+bx+c2n+1=-12n-1aax2+bx+c2n-1-b2adxax2+bx+c2n-1+C125. dxxax2+bx+c=-1cln2acx2+bcx+c2+bx+2cx+C126. dxxax2+bx+c=-1carcsinhbx+2cx4ac-b2+C127. sin2xdx=x2-sin2x4+C128. 1-sinxdx=cvsxdx=2cosx2+sinx2cosx2-sinx2，cvsx=21+sinx，其中cvsx是conversine函數(shù)129. sinnaxdx=-sinn-1axcosaxan+n-1nsinn-2

19、axdx+C 循環(huán)計(jì)算130. sinaxxdx=i=0-1iax2i+12i+12i+1!+C131. sinaxxndx=-sinaxn-1xn-1+an-1cosaxxn-1dx132. cosnaxdx=1ancosn-1axsinax+n-1ncosn-2axdx+C，n2133. cos2xdx=x2+sin2x4+C134. cosaxxdx=lnax+i-1-1iax2i2i2i!，n1135. cosaxxndx=-cosaxn-1xn-1-an-1sinaxxn-1dx，n1136. sinaxcosaxdx=12asin2ax137. sinaxsinbxdx=sina-

20、bx2a-b-sina+bx2a+b+C，a2b2138. sinaxcosbxdx=-cosa+bx2a+b-cosa-bx2a-b+C，a2b2139. cosaxcosbxdx=sina-bx2a-b+sina+bx2a+b+C，a2b2140. sinaxcosaxdx=-cos2ax4a+C，a0141. sinnaxcosaxdx=sinn+1axn+1a+C，a0，n-1142. cosnaxsinaxdx=-cosn+1axn+1a+C，a0，n-1143. tanaxdx=sinaxcosaxdx=-1alncosax+C，a0144. cotaxdx=cosaxsinaxd

21、x=1alnsinax+C，a0145. sinnaxcosmaxdx=-sinn-1axcosm+1axam+n+n-1m+nsinn-2axcosmaxdx+C=sinn+1axcosm-1axam+n+m-1n+msinnaxcosm-2axdx+C，a0，m+n0 循環(huán)計(jì)算146. sinaxsinbxdx=xsina-b2a-b-xsina+b2a+b+C，ab147. dxsinaxcosax=1alntanax+C148. dxsinaxcosnax=1an-1cosn-1ax+dxsinaxcosn-2ax，n1149. dxcosaxsinnax=-1an-1sinn-1ax

22、+dxcosaxsinn-2ax，n1150. sinaxdxcosnax=1an-1cosn-1ax+C，n1151. sin2axdxcosax=-1asinax+1alntan4+ax2+C152. sin2axdxcosnax=sinaxan-1cosn-1ax-1n-1dxcosn-2ax，n1153. sinnaxdxcosax=-sinn-1axan-1+sinn-2axdxcosax+C154. sinnaxdxcosmax=sinn+1axam-1cosm-1ax-n-m+2m-1sinnaxdxcosm-2ax+C=-sinn-1axan-mcosm-1ax+n-1n-ms

23、inn-2axdxcosmax+C=sinn-1axam-1cosm-1ax-n-1m-1sinn-1axdxcosm-2ax+C，m1，mn155. cosaxdxsinnax=-1an-1sinn-1ax+C156. cos2axdxsinax=1acosax+lntanax2+C157. cos2axdxsinnax=-1n-1cosaxasinn-1ax+dxsinn-2ax+C，n1158. cosnaxdxsinmax=-cosn+1axam-1sinm-1ax-n-m-2m-1cosnaxdxsinm-2ax+C=cosn-1axan-msinm-1ax+n-1n-mcosn-2

24、axdxsinmax+C=-cosn-1axam-1sinm-1ax-n-1m-1cosn-2axdxsinm-2ax+C，m1，mn159. dxb+csinax=-2ab2-c2arctanb-cb+ctan4-ax2+C，a0，b2>c2160. dxb+csinax=-1ac2-b2lnc+bsinax+c2-b2cosaxb+csinax+C，a0，b2<c2161. dx1+sinax=-1atan4-ax2+C，a0162. dx1-sinax=1atan4+ax2+C，a0163. x dx1+sinax=xatanax2-4+2c2lncosax2-4+C164.

25、 x dx1-sinax=xacot4-ax2+2c2lnsin4-ax2+C165. sinaxdx1±sinax=±x+1ctan4ax2+C166. dxb+ccosax=2ab2-c2arctanb-cb+ctanax2+C，a0，b2>c2167. dxb+ccosax=1ac2-b2lnc+bcosax+c2-b2sinaxb+ccosax+C，a0，b2<c2168. dx1+cosax=1atanax2+C，a0169. dx1-cosax=-1acotax2+C，a0170. x dx1+cosax=xatanax2+2a2lncosax2+C

26、，a0171. x dx1-cosax=-xacotax2+2a2lnsinax2+C，a0172. cosaxdx1+cosax=x-1atanax2+C173. cosaxdx1-cosax=-x-1acotax2+C174. cosaxcosbxdx=xsina-b2a-b+xsina+b2a+b+C，ab175. dxcosax±sinax=12alntanax2±8+C176. dxcosax+sinax2=12atanax4+C177. dxcosx+sinxn=1n-1sinx-cosxcosx+sinxn-1-2n-2dxcosx+sinxn-2+C178.

27、 dxcosax+sinaxn=179. cosaxdxcosax+sinax=x2+12alnsinax+cosax+C180. cosaxdxcosax-sinax=x2-12alnsinax-cosax+C181. sinaxdxcosax+sinax=x2-12alnsinax+cosax+C182. sinaxdxcosax-sinax=x2-12alnsinax-cosax+C183. cosaxdxsinax1+cosax=-14atan2ax2+12alntanax2+C184. cosaxdxsinax1-cosax=-14acot2ax2-12alntanax2+C185.

28、 sinaxdxcosax1+sinax=14acot2ax2+4+12alntanax2+4+C186. sinaxdxcosax1-sinax=14atan2ax2+4-12alntanax2+4+C187. sinaxtanaxdx=1alnsecax+tanax-sinax+C188. tannaxdxsin2ax=1an-1tann-1ax，n1189. tannaxdxcos2ax=1an+1tann+1ax，n-1190. cotnaxdxsin2ax=1an+1cotn+1ax，n-1191. cotnaxdxcos2ax=1a1-ntan1-nax，n1192. tanmax

29、cotnax=1am+n-1tanm+n-1ax-tanm-2axcotnaxdx，m+n1193. xsinaxdx=1a2sinax-xacosax+C，a0194. xcosaxdx=cosaxa2+xsinaxa+C195. xnsinaxdx=-xnacosax+naxn-1cosaxdx，a0 循環(huán)計(jì)算196. xncosaxdx=xnasinax-naxn-1sinaxdx，a0 循環(huán)計(jì)算197. tanaxdx=-1alncosax+C，a0198. cotaxdx=1alnsinax+C，a0199. tan2axdx=1atanax-x+C，a0200. cot2axdx=

30、-1acotax-x+C，a0201. tannaxdx=tann-1axan-1-tann-2axdx，a0，n1 循環(huán)計(jì)算202. cotnaxdx=-cotn-1axan-1-cotn-2axdx，a0，n1 循環(huán)計(jì)算203. dxtanax+1=x2+12alnsinax+cosax+C204. dxtanax-1=-x2+12alnsinax-cosax+C205. tanaxdxtanax+1=x2-12alnsinax+cosax+C206. tanaxdxtanax-1=x2+12alnsinax-cosax+C207. dx1+cotax=tanaxdxtanax+1=x2-

31、12alnsinax+cosax+C208. dx1-cotax=tanaxdxtanax-1=x2+12alnsinax-cosax+C209. secaxdx=1alnsecax+tanax+C=1alntanax2+4，a0210. cscaxdx=-1alncscax+cotax+C=1alntanax2+C，a0211. secnaxdx=secn-2axtanaxan-1+n-2n-1secn-2axdx，a0，n1 循環(huán)計(jì)算212. cscnaxdx=-cscn-2axcotaxan-1+n-2n-1cscn-2axdx，a0，n1 循環(huán)計(jì)算213. secnaxtanaxdx=

32、secnaxna+C，a0，n0214. cscnaxcotaxdx=-cscnaxna+C，a0，n0215. dxsecx+1=x-tanx2+C216. arcsinaxdx=xarcsinax+1a1-a2x2+C，a0217. xarcsinxadx=x22-a24arcsinxa+x4c2-x2+C218. x2arcsinxadx=x33arcsinxa+x2+2c29c2-x2+C219. xnarcsinxdx=1n+1xn+1arcsinx+xn1-x2-nxn-1arcsinxn-1+nxn-2arcsinxdx+C220. arccosaxdx=xarccosax-1a

33、1-a2x2+C，a0221. xarccosxadx=x22-a24arccosxa-x4a2-x2+C222. x2arccosxadx=x33arccosxa-x2+2a29a2-x2+C223. arctanaxdx=xarctanax-12aln1+a2x2+C，a0224. xarctanxadx=a2+x22arctanxa-ax2+C225. x2arctanxadx=x33arctanxa-ax26+a36lna2+x2+C226. xnarctanxadx=xn+1n+1arctanxa-an+1xn+1a2+x2dx+C，n1227. arccotaxdx=xarccot

34、ax+12aln1+a2x2+C228. xarccotxadx=a2+x22arccotxa+ax2+C229. x2arccotxadx=x33arccotxa+ax26-a36lna2+x2+C230. xnarccotxadx=xn+1n+1arccotxa+an+1xn+1a2+x2dx，n1231. arcsecaxdx=xarcsecax+axxlnx±x2-1+C232. xarcsecxdx=12x2arcsecx-x2-1+C233. xnarcsecxdx=1n+1xn+1arcsecx-1nxn-1x2-1+1-nxn-1arcsecx+1-nxn-2arcs

35、ecxdx+C234. arccscaxdx=xarccscax-axxlnx±x2-1+C235. sinhaxdx=1acoshax+C236. coshaxdx=1asinhax+C237. sinh2axdx=14asinh2ax-x2+C238. cosh2axdx=14asinh2ax+x2+C239. sinhnaxdx=1ansinhn-1axcoshax-n-1nsinhn-2axdx+C，n>0 =1an+1sinhn+1axcoshax-n+2n+1sinhn+2axdx+C，n<0,n-1240. coshnaxdx=1ansinhaxcoshn-

36、1ax+n-1ncoshn+2axdx，n<0,n-1241. dxsinhax=1alntanhax2+C=1alncoshax-1sinhax+C=1alnsinhaxcoshax+1+C=1alncoshax-1coshax+1+C242. dxcoshax=2aarctaneax+C243. dxsinhnax=coshaxan-1sinhn-1ax-n-2n-1dxsinhn-2ax，n1244. dxcoshnax=sinhaxan-1coshn-1ax+n-2n-1dxcoshn-2ax，n1245. coshnaxsinhmaxdx=coshn-1axan-msinhm-

37、1ax+n-1n-mcoshn-2axsinhmaxdx=-coshn+1axam-1sinhm-1ax+n-m+2m-1coshnaxsinhm-2axdx+C=-coshn-1axam-1sinhm-1ax+n-1m-1coshn-2axsinhm-2axdx+C，mn,m1246. sinhmaxcosnaxdx=sinhm-1axam-ncoshn-1ax+m-1m-nsinhm-2axcoshnaxdx+C=sinhm+1axan-1coshn-1ax+m-n+2n-1sinhmaxcoshn-2axdx+C=sinhm-1axan-1coshn-1ax+m-1n-1sinhm-2a

38、xcoshn-2axdx+C，mn,n1247. xsinhaxdx=1axcoshax-1a2sinhax+C248. xcoshaxdx=1axsinhax-1a2coshax+C249. tanhaxdx=1alncoshax+C250. cothaxdx=1alnsinhax+C251. tanhnaxdx=-tanhn-1axan-1+tanhn-2axdx+C，n1252. cothnaxdx=-cothn-1axan-1+cothn-2axdx，n1253. sinhaxsinhbxdx=asinhbxcoshax-bcoshbxsinhaxa2-b2+C254. coshaxc

39、osbxdx=asinhaxcoshbx-bsinhbxcoshaxa2-b2+C255. coshaxsinhbxdx=asinhaxsinhbx-bcoshaxcoshbxa2-b2+C256. sinhax+bsincx+ddx=aa2+c2coshax+bsincx+d-ca2+c2sinhax+bcoscx+d+C257. sinhax+bcoscx+ddx=aa2+c2coshax+bcoscx+d+ca2+c2sinhax+bsincx+d+C258. coshax+bsincx+ddx=aa2+c2sinhax+bsincx+d-ca2+c2coshax+bcoscx+d+C2

40、59. coshax+bcoscx+ddx=aa2+c2sinhax+bcoscx+d+ca2+c2coshax+bsincx+d+C260. arcsinhxadx=xarcsinhxa-x2+a2+C261. arccoshxadx=xarccoshxa-x2-a2+C262. arctanhxadx=xarctanhxa+a2lna2-x2+C，x<a263. arccothxadx=xarccothxa+a2lnx2-a2+C，x<a264. arcsechxadx=xarcsechxa-aarctanxa-xa+xx-a+C，x0,a265. arccschxadx=xarccschxa+alnx+x2+a2a+