常用函数积分表(增强)

1、1. sec2xdx=tanx+C2. csc2xdx=-cotx+C3. secxtanxdx=secx+C4. cscxcotxdx=-cscx+C5. xax+bndx=ax+bn+1a2ax+bn+2-bn+1+C，a0，n-1,-26. xax+bdx=xa-ba2lnax+b+C，a07. xax+b2dx=1a2lnax+b+bax+b+C，a08. x2ax+bdx=12a3ax+b2-4bax+b+2b2lnax+b+C9. x2ax+b2dx=1a3ax+b-2blnax+b-b2ax+b+C10. x2ax+b3dx=1a3lnax+b+2bax+b-b22ax+b2+C

2、11. x2ax+bndx=1a3-1n-3ax+bn-3+2bn-2ax+bn-2-b2n-1ax+bn-1+C，n1,2,312. dxxax+b=1blnxax+b+C，b013. dxx2ax+b=-1bx+ab2lnax+bx+C14. dxx2ax+b2=-a1b2ax+b+1ab2x-2b3lnax+bx+C15. xax+bdx=215a23ax-2bax+b32+C16. x2ax+bdx=2105a315a2x2-12abx+8b2ax+b32+C17. ax+bndx=2ax+bn+2an+2+C，a0，n-218. xnax+bdx=2a2n+3xnax+b32-2nb

3、a2n+3xn-1ax+bdx循环计算19. ax+bxdx=2ax+b+bdxxax+b=2ax+b-2barctanhax+bb+C1 / 1920. dxxax+b=2-barctanax+b-b+C，b<021. dxxax+b=1blnax+b-bax+b+b+C，b>022. ax+bx2dx=-ax+bx+a2dxxax+b+C23. ax+bxndx=-ax+b32bn-1xn-1-2n-5a2bn-1ax+bxn-1dx，n1循环计算24. xnax+bdx=2a2n+1xnax+b-bnxn-1ax+b+C循环计算25. dxx2ax+b=-ax+bbx-a2b

4、dxxax+b+C，b026. dxxnax+b=-ax+bbn-1xn-1-2n-3a2bn-1ax+bxn-1dx，n1循环计算27. xnax+bdx=22n+1xn+1ax+b+bxnax+b-nbxn-1ax+bdx+C循环计算28. dxa2+x2=1aarctanxa+C，a029. dxa2+x22=x2a2a2+x2+12a3arctanxa+C，a030. dxa2-x2=12alna+xa-x+C=1aarctanhxa+C，a0，a>x31. dxa2-x22=x2a2a2-x2+14a3lnx+ax-a+C32. 1x2-a2dx=12alnx-ax+a+C=-

5、1aarccothxa+C，a0，x>a33. dxa2+x2=lnx+a2+x2+C34. a2+x2dx=x2a2+x2+a22lnx+a2+x2+C35. a2+x23dx=xa2+x234+38a2xa2+x2+38a4lnx+a2+x2+C36. a2+x25dx=xa2+x256+524a2xa2+x23+516a4xa2+x2+516a6lnx+a2+x2+C37. xa2+x22n+1dx=a2+x22n+32n+3+C38. x2a2+x2dx=x8a2+2x2a2+x2-a48lnx+a2+x2+C39. x2a2+x23dx=xa2+x256-a2xa2+x224-

6、a4xa2+x216-a616lnx+a2+x2+C40. x3a2+x2dx=a2+x255-a2a2+x233+C41. x3a2+x23dx=a2+x277-a2a2+x255+C42. x3a2+x22n+1dx=a2+x22n+52n+5-a2a2+x22n+32n+3+C43. x4a2+x2dx=x3a2+x236-a2xa2+x238+a4xa2+x216+a616lnx+a2+x2+C44. x4a2+x23dx=x3a2+x258-a2xa2+x2516+a4xa2+x2364+3a6xa2+x2128+3a8128lnx+a2+x2+C45. x5a2+x2dx=a2+x

7、277-2a2a2+x255+a4a2+x233+C46. x5a2+x23dx=a2+x299-2a2a2+x277+a4a2+x255+C47. x5a2+x22n+1dx=a2+x22n+72n+7-2a2a2+x22n+52n+5+a4a2+x22n+32n+3+C48. a2+x2xdx=a2+x2-alna+a2+x2x+C=a2+x2-aarcsinhax+C49. a2+x23xdx=a2+x233+a2a2+x2-a3lna+a2+x2x+C50. a2+x25xdx=a2+x255+a2a2+x233+a4a2+x2-a5lna+a2+x2x+C51. a2+x277dx=

8、a2+x277+a2a2+x255+a4a2+x233+a6a2+x2-a7lna+a2+x2x+C52. a2+x2x2dx=lnx+a2+x2-a2+x2x+C53. x2a2+x2dx=-a22lnx+a2+x2+xa2+x22+C=-a22arcsinhxa+xa2+x22+C54. dxxa2+x2=-1alna+a2+x2x+C=-1aarcsinhax+C55. dxx2a2+x2=-a2+x2a2x+C，a056. dxa2-x2=arcsinxa+C，a0，xa57. a2-x2dx=x2a2-x2+a22arcsinxa+C，a0，xa58. a2-x2dx=12xa2-x

9、2-sgnxarccoshxa+C，xa59. xa2-x2dx=-a2-x233+C，xa60. x2a2-x2dx=a48arcsinxa-18xa2-x2a2-2x2+C，a061. a2-x2xdx=a2-x2-alna+a2-x2x+C，xa62. a2-x2x2dx=-arcsinxa-a2-x2x+C，a063. x2a2-x2dx=a22arcsinxa-xa2-x22+C，a0，xa2-x264. dxxa2-x2=-1alna+a2-x2x+C，a065. dxx2a2-x2=-a2-x2a2x+C，a066. dxx2-a2=lnx+x2-a2+C67. x2-a2dx=

10、x2x2-a2-a22lnx+x2-a2+C68. x2-a2ndx=xx2-a2nn+1-na2n+1x2-a2n-2dx，n-1循环计算69. dxx2-a2n=xx2-a22-n2-na2+n-32-na2dxx2-a2n-2，n2循环计算70. xx2-a2ndx=x2-a2n+2n+2+C，n-271. x2x2-a2dx=x82x2-a2x2-a2-a48lnx+x2-a2+C72. x2-a2xdx=x2-a2-aarcsecxa+C=x2-a2-aarccosax+C，a073. xdxx2-a2=x2-a2+C74. xdxx2-a23=-1x2-a2+C75. xdxx2-

11、a25=-13x2-a23+C76. xdxx2-a27=-15x2-a25+C77. xdxx2-a22n+1=-12n-1x2-a22n-1+C78. x2-a2x2dx=lnx+x2-a2-x2-a2x+C79. x2x2-a2dx=a22lnx+x2-a2+x2x2-a2+C80. x2x2-a23dx=-xx2-a2+lnx+x2-a2a+C81. x4x2-a2dx=x3x2-a24+38a2xx2-a2+38a4lnx+x2-a2a+C82. x4x2-a23dx=xx2-a22-a2xx2-a2+32a2lnx+x2-a2a+C83. x4x2-a25dx=-xx2-a2-x3

12、3x2-a23+lnx+x2-a2a+C84. x2mdxx2-a22n+1=-x2m-12n-1x2-a22n-1+2m-12n-1x2m-2x2-a22n-1dx+C=-1n-ma2n-mi=0n-m-112m+i+1n-m-1ix2m+i+1x2-a22m+i+1，n>m085. dxx2-a23=-xa2x2-a2+C86. dxx2-a25=1a4xx2-a2-x33x2-a23+C87. dxx2-a27=-1a6xx2-a2-2x33x2-a23+x55x2-a25+C88. dxx2-a29=1a8xx2-a2-2x33x2-a23+3x55x2-a25-x77x2-a2

13、7+C89. x2x2-a25dx=-x33a2x2-a23+C90. x2x2-a27dx=1a4x33x2-a23-x55x2-a25+C91. x2x2-a29dx=-1a6x33x2-a23-2x55x2-a25+x77x2-a27+C92. dxxx2-a2=1aarcsecxa+C，a093. dxx2x2-a2=x2-a2a2x+C，a094. dxax2+bx+c=24ac-b2arctan2ax+b4ac-b2，4ac-b2>095. dxax2+bx+c=2b2-4acarctanh2ax+bb2-4ac=1b2-4acln2ax+b-b2-4ac2ax+b+b2-4

14、ac，4ac-b2<096. dxax2+bx+c=-22ax+b，4ac-b2=097. dxax2+bx+c=12alnax2+bx+c-b2adxax2+bx+c+C98. mx+nax2+bx+cdx=m2alnax2+bx+c+2an-bma4ac-b2arctan2ax+b4ac-b2+C，4ac-b2>099. mx+nax2+bx+cdx=m2alnax2+bx+c+2an-bmab2-4acarctanh2ax+bb2-4ac+C，4ac-b2<0100. mx+nax2+bx+cdx=m2alnax2+bx+c-2an-bma2ax+b+C，4ac-b2=

15、0101. dxax2+bx+cn=2ax+bn-14ac-b2ax2+bx+cn-1+2n-32an-14ac-b2dxax2+bx+cn-1+C102. xax2+bx+cndx=bx+2cn-14ac-b2ax2+bx+cn-1-b2n-3n-14ac-b2dxax2+bx+cn-1+C103. dxxax2+bx+c=12clnx2ax2+bx+c-b2cdxax2+bx+c+C104. dxax2+bx+c=1aln2a2x2+abx+ac+2ax+b+C，a>0105. dxax2+bx+c=1aarcsinh2ax+b4ac-b2+C，a>0，4ac-b2>01

16、06. dxax2+bx+c=1aln2ax+b+C，a>0，4ac-b2=0107. dxax2+bx+c=-1-aarcsin2ax+bb2-4ac+C，a<0，4ac-b2<0108. dxax2+bx+c3=4ax+2b4ac-b2ax2+bx+c+C109. dxax2+bx+c5=4ax+2b34ac-b2ax2+bx+c1ax2+bx+c+8a4ac-b2+C110. dxax2+bx+c2n+1=4ax+2b2n-14ac-b2ax2+bx+c2n-1+8an-12n-14ac-b2dxax2+bx+c2n-1+C循环计算111. xdxax2+bx+c=ax

17、2+bx+ca-b2adxax2+bx+c+C112. xdxax2+bx+c3=-2bx+4c4ac-b2ax2+bx+c+C113. xdxax2+bx+c2n+1=-12n-1aax2+bx+c2n-1-b2adxax2+bx+c2n-1+C114. dxxax2+bx+c=-1cln2acx2+bcx+c2+bx+2cx+C115. dxxax2+bx+c=-1carcsinhbx+2cx4ac-b2+C116. sin2xdx=x2-sin2x4+C117. 1-sinxdx=cvsxdx=2cosx2+sinx2cosx2-sinx2，cvsx=21+sinx，其中cvsx是con

18、versine函数118. sinnaxdx=-sinn-1axcosaxan+n-1nsinn-2axdx+C循环计算119. sinaxxdx=i=0-1iax2i+12i+12i+1!+C120. sinaxxndx=-sinaxn-1xn-1+an-1cosaxxn-1dx121. cosnaxdx=1ancosn-1axsinax+n-1ncosn-2axdx+C，n2122. cos2xdx=x2+sin2x4+C123. cosaxxdx=lnax+i-1-1iax2i2i2i!，n1124. cosaxxndx=-cosaxn-1xn-1-an-1sinaxxn-1dx，n11

19、25. sinaxcosaxdx=12asin2ax126. sinaxsinbxdx=sina-bx2a-b-sina+bx2a+b+C，a2b2127. sinaxcosbxdx=-cosa+bx2a+b-cosa-bx2a-b+C，a2b2128. cosaxcosbxdx=sina-bx2a-b+sina+bx2a+b+C，a2b2129. sinaxcosaxdx=-cos2ax4a+C，a0130. sinnaxcosaxdx=sinn+1axn+1a+C，a0，n-1131. cosnaxsinaxdx=-cosn+1axn+1a+C，a0，n-1132. tanaxdx=sin

20、axcosaxdx=-1alncosax+C，a0133. cotaxdx=cosaxsinaxdx=1alnsinax+C，a0134. sinnaxcosmaxdx=-sinn-1axcosm+1axam+n+n-1m+nsinn-2axcosmaxdx+C=sinn+1axcosm-1axam+n+m-1n+msinnaxcosm-2axdx+C，a0，m+n0循环计算135. sinaxsinbxdx=xsina-b2a-b-xsina+b2a+b+C，ab136. dxsinaxcosax=1alntanax+C137. dxsinaxcosnax=1an-1cosn-1ax+dxs

21、inaxcosn-2ax，n1138. dxcosaxsinnax=-1an-1sinn-1ax+dxcosaxsinn-2ax，n1139. sinaxdxcosnax=1an-1cosn-1ax+C，n1140. sin2axdxcosax=-1asinax+1alntan4+ax2+C141. sin2axdxcosnax=sinaxan-1cosn-1ax-1n-1dxcosn-2ax，n1142. sinnaxdxcosax=-sinn-1axan-1+sinn-2axdxcosax+C143. sinnaxdxcosmax=sinn+1axam-1cosm-1ax-n-m+2m-1

22、sinnaxdxcosm-2ax+C=-sinn-1axan-mcosm-1ax+n-1n-msinn-2axdxcosmax+C=sinn-1axam-1cosm-1ax-n-1m-1sinn-1axdxcosm-2ax+C，m1，mn144. cosaxdxsinnax=-1an-1sinn-1ax+C145. cos2axdxsinax=1acosax+lntanax2+C146. cos2axdxsinnax=-1n-1cosaxasinn-1ax+dxsinn-2ax+C，n1147. cosnaxdxsinmax=-cosn+1axam-1sinm-1ax-n-m-2m-1cosn

23、axdxsinm-2ax+C=cosn-1axan-msinm-1ax+n-1n-mcosn-2axdxsinmax+C=-cosn-1axam-1sinm-1ax-n-1m-1cosn-2axdxsinm-2ax+C，m1，mn148. dxb+csinax=-2ab2-c2arctanb-cb+ctan4-ax2+C，a0，b2>c2149. dxb+csinax=-1ac2-b2lnc+bsinax+c2-b2cosaxb+csinax+C，a0，b2<c2150. dx1+sinax=-1atan4-ax2+C，a0151. dx1-sinax=1atan4+ax2+C，a

24、0152. xdx1+sinax=xatanax2-4+2c2lncosax2-4+C153. xdx1-sinax=xacot4-ax2+2c2lnsin4-ax2+C154. sinaxdx1±sinax=±x+1ctan4ax2+C155. dxb+ccosax=2ab2-c2arctanb-cb+ctanax2+C，a0，b2>c2156. dxb+ccosax=1ac2-b2lnc+bcosax+c2-b2sinaxb+ccosax+C，a0，b2<c2157. dx1+cosax=1atanax2+C，a0158. dx1-cosax=-1acota

25、x2+C，a0159. xdx1+cosax=xatanax2+2a2lncosax2+C，a0160. xdx1-cosax=-xacotax2+2a2lnsinax2+C，a0161. cosaxdx1+cosax=x-1atanax2+C162. cosaxdx1-cosax=-x-1acotax2+C163. cosaxcosbxdx=xsina-b2a-b+xsina+b2a+b+C，ab164. dxcosax±sinax=12alntanax2±8+C165. dxcosax+sinax2=12atanax4+C166. dxcosx+sinxn=1n-1si

26、nx-cosxcosx+sinxn-1-2n-2dxcosx+sinxn-2+C167. dxcosax+sinaxn=168. cosaxdxcosax+sinax=x2+12alnsinax+cosax+C169. cosaxdxcosax-sinax=x2-12alnsinax-cosax+C170. sinaxdxcosax+sinax=x2-12alnsinax+cosax+C171. sinaxdxcosax-sinax=x2-12alnsinax-cosax+C172. cosaxdxsinax1+cosax=-14atan2ax2+12alntanax2+C173. cosax

27、dxsinax1-cosax=-14acot2ax2-12alntanax2+C174. sinaxdxcosax1+sinax=14acot2ax2+4+12alntanax2+4+C175. sinaxdxcosax1-sinax=14atan2ax2+4-12alntanax2+4+C176. sinaxtanaxdx=1alnsecax+tanax-sinax+C177. tannaxdxsin2ax=1an-1tann-1ax，n1178. tannaxdxcos2ax=1an+1tann+1ax，n-1179. cotnaxdxsin2ax=1an+1cotn+1ax，n-1180

28、. cotnaxdxcos2ax=1a1-ntan1-nax，n1181. tanmaxcotnax=1am+n-1tanm+n-1ax-tanm-2axcotnaxdx，m+n1182. xsinaxdx=1a2sinax-xacosax+C，a0183. xcosaxdx=cosaxa2+xsinaxa+C184. xnsinaxdx=-xnacosax+naxn-1cosaxdx，a0循环计算185. xncosaxdx=xnasinax-naxn-1sinaxdx，a0循环计算186. tanaxdx=-1alncosax+C，a0187. cotaxdx=1alnsinax+C，a0

29、188. tan2axdx=1atanax-x+C，a0189. cot2axdx=-1acotax-x+C，a0190. tannaxdx=tann-1axan-1-tann-2axdx，a0，n1循环计算191. cotnaxdx=-cotn-1axan-1-cotn-2axdx，a0，n1循环计算192. dxtanax+1=x2+12alnsinax+cosax+C193. dxtanax-1=-x2+12alnsinax-cosax+C194. tanaxdxtanax+1=x2-12alnsinax+cosax+C195. tanaxdxtanax-1=x2+12alnsinax-

30、cosax+C196. dx1+cotax=tanaxdxtanax+1=x2-12alnsinax+cosax+C197. dx1-cotax=tanaxdxtanax-1=x2+12alnsinax-cosax+C198. secaxdx=1alnsecax+tanax+C=1alntanax2+4，a0199. cscaxdx=-1alncscax+cotax+C=1alntanax2+C，a0200. secnaxdx=secn-2axtanaxan-1+n-2n-1secn-2axdx，a0，n1循环计算201. cscnaxdx=-cscn-2axcotaxan-1+n-2n-1c

31、scn-2axdx，a0，n1循环计算202. secnaxtanaxdx=secnaxna+C，a0，n0203. cscnaxcotaxdx=-cscnaxna+C，a0，n0204. dxsecx+1=x-tanx2+C205. arcsinaxdx=xarcsinax+1a1-a2x2+C，a0206. xarcsinxadx=x22-a24arcsinxa+x4c2-x2+C207. x2arcsinxadx=x33arcsinxa+x2+2c29c2-x2+C208. xnarcsinxdx=1n+1xn+1arcsinx+xn1-x2-nxn-1arcsinxn-1+nxn-2a

32、rcsinxdx+C209. arccosaxdx=xarccosax-1a1-a2x2+C，a0210. xarccosxadx=x22-a24arccosxa-x4a2-x2+C211. x2arccosxadx=x33arccosxa-x2+2a29a2-x2+C212. arctanaxdx=xarctanax-12aln1+a2x2+C，a0213. xarctanxadx=a2+x22arctanxa-ax2+C214. x2arctanxadx=x33arctanxa-ax26+a36lna2+x2+C215. xnarctanxadx=xn+1n+1arctanxa-an+1x

33、n+1a2+x2dx+C，n1216. arccotaxdx=xarccotax+12aln1+a2x2+C217. xarccotxadx=a2+x22arccotxa+ax2+C218. x2arccotxadx=x33arccotxa+ax26-a36lna2+x2+C219. xnarccotxadx=xn+1n+1arccotxa+an+1xn+1a2+x2dx，n1220. arcsecaxdx=xarcsecax+axxlnx±x2-1+C221. xarcsecxdx=12x2arcsecx-x2-1+C222. xnarcsecxdx=1n+1xn+1arcsecx

34、-1nxn-1x2-1+1-nxn-1arcsecx+1-nxn-2arcsecxdx+C223. arccscaxdx=xarccscax-axxlnx±x2-1+C224. sinhaxdx=1acoshax+C225. coshaxdx=1asinhax+C226. sinh2axdx=14asinh2ax-x2+C227. cosh2axdx=14asinh2ax+x2+C228. sinhnaxdx=1ansinhn-1axcoshax-n-1nsinhn-2axdx+C，n>0=1an+1sinhn+1axcoshax-n+2n+1sinhn+2axdx+C，n&l

35、t;0,n-1229. coshnaxdx=1ansinhaxcoshn-1ax+n-1ncoshn+2axdx，n<0,n-1230. dxsinhax=1alntanhax2+C=1alncoshax-1sinhax+C=1alnsinhaxcoshax+1+C=1alncoshax-1coshax+1+C231. dxcoshax=2aarctaneax+C232. dxsinhnax=coshaxan-1sinhn-1ax-n-2n-1dxsinhn-2ax，n1233. dxcoshnax=sinhaxan-1coshn-1ax+n-2n-1dxcoshn-2ax，n1234.

36、 coshnaxsinhmaxdx=coshn-1axan-msinhm-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,m1235. sinhmaxcosnaxdx=sinhm-1axam-ncoshn-1ax+m-1m-nsinhm-2axcoshnaxdx+C=sinhm+1axan-1coshn-1ax+m-n+2n-1sinhmaxcoshn-2axdx+C=

37、sinhm-1axan-1coshn-1ax+m-1n-1sinhm-2axcoshn-2axdx+C，mn,n1236. xsinhaxdx=1axcoshax-1a2sinhax+C237. xcoshaxdx=1axsinhax-1a2coshax+C238. tanhaxdx=1alncoshax+C239. cothaxdx=1alnsinhax+C240. tanhnaxdx=-tanhn-1axan-1+tanhn-2axdx+C，n1241. cothnaxdx=-cothn-1axan-1+cothn-2axdx，n1242. sinhaxsinhbxdx=asinhbxco

38、shax-bcoshbxsinhaxa2-b2+C243. coshaxcosbxdx=asinhaxcoshbx-bsinhbxcoshaxa2-b2+C244. coshaxsinhbxdx=asinhaxsinhbx-bcoshaxcoshbxa2-b2+C245. sinhax+bsincx+ddx=aa2+c2coshax+bsincx+d-ca2+c2sinhax+bcoscx+d+C246. sinhax+bcoscx+ddx=aa2+c2coshax+bcoscx+d+ca2+c2sinhax+bsincx+d+C247. coshax+bsincx+ddx=aa2+c2sin

39、hax+bsincx+d-ca2+c2coshax+bcoscx+d+C248. coshax+bcoscx+ddx=aa2+c2sinhax+bcoscx+d+ca2+c2coshax+bsincx+d+C249. arcsinhxadx=xarcsinhxa-x2+a2+C250. arccoshxadx=xarccoshxa-x2-a2+C251. arctanhxadx=xarctanhxa+a2lna2-x2+C，x<a252. arccothxadx=xarccothxa+a2lnx2-a2+C，x<a253. arcsechxadx=xarcsechxa-aarctanxa-xa+xx-a+C，x0,a254. arccschxadx=xarccschxa+alnx+x2+a2a+C，x0,a255. xeaxdx=eaxa2ax-1+C，a0256. baxdx=baxalnb+C，a0，b>0，b1257.