常用函数积分表(增强)

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+C11. 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+b32-2nba2n+3xn-1ax+bdx循环计算19. ax+bxdx=2ax+b+bdxxax+b=2ax+b-2barctanhax+bb+C20. dxxax+b=2-barctanax+b-b+C，b022. 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-a2bdxxax+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，ax31. dxa2-x22=x2a2a2-x2+14a3lnx+ax-a+C32. 1x2-a2dx=12alnx-ax+a+C=-1aarccothxa+C，a0，xa33. 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-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+x277-2a2a2+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=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-x2-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=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-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-x33x2-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，nm085. 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-a27+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-b2095. dxax2+bx+c=2b2-4acarctanh2ax+bb2-4ac=1b2-4acln2ax+b-b2-4ac2ax+b+b2-4ac，4ac-b2099. mx+nax2+bx+cdx=m2alnax2+bx+c+2an-bmab2-4acarctanh2ax+bb2-4ac+C，4ac-b20105. dxax2+bx+c=1aarcsinh2ax+b4ac-b2+C，a0，4ac-b20106. dxax2+bx+c=1aln2ax+b+C，a0，4ac-b2=0107. dxax2+bx+c=-1-aarcsin2ax+bb2-4ac+C，a0，4ac-b2c2149. dxb+csinax=-1ac2-b2lnc+bsinax+c2-b2cosaxb+csinax+C，a0，b2c2156. dxb+ccosax=1ac2-b2lnc+bcosax+c2-b2sinaxb+ccosax+C，a0，b20=1an+1sinhn+1axcoshax-n+2n+1sinhn+2axdx+C，n0,n-1229. coshnaxdx=1ansinhaxcoshn-1ax+n-1ncoshn+2axdx，n0,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. 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=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=asinhbxcoshax-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+c2sinhax+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，xa252. arccothxadx=xarccothxa+a2lnx2-a2+C，x0，b1257. x2eaxdx=eaxx2a-2xa2+2a3+C258. xneaxdx=xneaxa-naxn-1eaxdx，a0259. eaxdxx=lnx+i=1axiii!+C260. eaxdxxn=1n-1-eaxxn-1+aeaxxn-1dx+C，n1261. eaxlnxdx=1aeaxlnx-Eiax+C262. eaxsinbxdx=eaxa2+b2asinbx-bcosbx+C263. eaxcosbxdx=eaxa2+b2acosbx+bsinbx+C264. eaxsinnbxdx=eaxsinn-1xa2+n2asinx-ncosx+nn-1a2+n2eaxsinn-2xdx265. eaxcosnbxdx=eaxcosn-1xa2+n2acosx+nsinx+nn-1a2+n2eaxcosn-2xdx266. xeax2dx=12aeax2+C267. 12e-x-222dx=121+erfx-2+C268. ex2dx=ex2j=0n-1a2jx2j+1+2n-1a2n-2ex2x2ndx，n0其中a2j=1352j-12j+1=2j!j!22j+1+C269. -e-ax2dx=a the Gaussian integral270. 0x2ne-x2a2dx=2n!n!a22n+1271. lnaxdx=xlnax-x+C272. lnx2dx=xlnx2-2xlnx+2x+C273. lnaxndx=xlnaxn-nlnaxn-1dx274. dxlnx=lnlnx+lnx+i=2lnxiii!+C275. dxlnxn=-xn-1lnxn-1+1n-1dxlnxn-1+C，n1276. xmlnxdx=xm+1lnxm+1-1m+12+C，n-1277. xmlnxndx=xm+1lnxnm+1-nm+1xmlnxn-1dx+C，m-1278. lnxndxx=lnxn+1n+1+C，n-1279. lnxdxxm=-lnxm-1xm-1-1m-12xm-1，m1280. lnxndxxm=-lnxnm-1xm-1+nm-1lnxn-1dxxm，m1281. xmdxlnxn=-xm+1n-1lnxn-1+m+1n-1xmdxlnxn-1，n1282. xnlnaxmdx=xn+1lnaxmn+1-mn+1xnlnaxm-1dx，n-1283. lnaxmxdx=lnaxm+1m+1+C，m-1284. dxxlnax=lnlnax+C285. dxxnlnx=lnlnx+i=1-1in-1ilnxiii!+C286. dxxlnxn=-1n-1lnxn-1，n1287. sinlnxdx=x2sinlnx-coslnx+C288. coslnxdx=x2sinlnx+coslnx+C289. exxlnx-x-1xdx=exxlnx-x-lnx+C