常用函数积分表(增强版)

1. fx+gxdx=fxdx+gxdx2. fx-gxdx=fxdx-gxdx3. fxdgx=fxgx-gxdfx4. axdx=axlna+C，a1，a05. 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，a016. 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. dxxax+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 循环计算30. ax+bxdx=2ax+b+bdxxax+b=2ax+b-2barctanhax+bb+C31. dxxax+b=2-barctanax+b-b+C，b033. ax+bx2dx=-ax+bx+a2dxxax+b+C34. ax+bxndx=-ax+b32bn-1xn-1-2n-5a2bn-1ax+bxn-1dx，n1 循环计算35. xnax+bdx=2a2n+1xnax+b-bnxn-1ax+b+C 循环计算36. dxx2ax+b=-ax+bbx-a2bdxxax+b+C，b037. dxxnax+b=-ax+bbn-1xn-1-2n-3a2bn-1ax+bxn-1dx，n1 循环计算38. xnax+bdx=22n+1xn+1ax+b+bxnax+b-nbxn-1ax+bdx+C 循环计算39. dxa2+x2=1aarctanxa+C，a040. dxa2+x22=x2a2a2+x2+12a3arctanxa+C，a041. dxa2-x2=12alna+xa-x+C=1aarctanhxa+C，a0，ax42. dxa2-x22=x2a2a2-x2+14a3lnx+ax-a+C43. 1x2-a2dx=12alnx-ax+a+C=-1aarccothxa+C，a0，xa44. dxa2+x2=lnx+a2+x2+C45. a2+x2dx=x2a2+x2+a22lnx+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=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. 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-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-18xa2-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 循环计算80. dxx2-a2n=xx2-a22-n2-na2+n-32-na2dxx2-a2n-2，n2 循环计算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-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-1x2m-2x2-a22n-1dx+C=-1n-ma2n-mi=0n-m-112m+i+1n-m-1ix2m+i+1x2-a22m+i+1，nm096. 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-x55x2-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-b20106. dxax2+bx+c=2b2-4acarctanh2ax+bb2-4ac=1b2-4acln2ax+b-b2-4ac2ax+b+b2-4ac，4ac-b20110. mx+nax2+bx+cdx=m2alnax2+bx+c+2an-bmab2-4acarctanh2ax+bb2-4ac+C，4ac-b20116. dxax2+bx+c=1aarcsinh2ax+b4ac-b2+C，a0，4ac-b20117. dxax2+bx+c=1aln2ax+b+C，a0，4ac-b2=0118. dxax2+bx+c=-1-aarcsin2ax+bb2-4ac+C，a0，4ac-b2c2160. dxb+csinax=-1ac2-b2lnc+bsinax+c2-b2cosaxb+csinax+C，a0，b2c2167. dxb+ccosax=1ac2-b2lnc+bcosax+c2-b2sinaxb+ccosax+C，a0，b20 =1an+1sinhn+1axcoshax-n+2n+1sinhn+2axdx+C，n0,n-1240. coshnaxdx=1ansinhaxcoshn-1ax+n-1ncoshn+2axdx，n0,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-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-2axcoshn-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. coshaxcosbxdx=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+C259. 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，xa263. arccothxadx=xarccothxa+a2lnx2-a2+C，x0，b1268. x2eaxdx=eaxx2a-2xa2+2a3+C269. xneaxdx=xneaxa-naxn-1eaxdx，a0270. eaxdxx=lnx+i=1axiii!+C271. eaxdxxn=1n-1-eaxxn-1+aeaxxn-1dx+C，n1272. eaxlnxdx=1aeaxlnx-Eiax+C273. eaxsinbxdx=eaxa2+b2asinbx-bcosbx+C274. eaxcosbxdx=eaxa2+b2acosbx+bsinbx+C275. eaxsinnbxdx=eaxsinn-1xa2+n2asinx-ncosx+nn-1a2+n2eaxsinn-2xdx276. eaxcosnbxdx=eaxcosn-1xa2+n2acosx+nsinx+nn-1a2+n2eaxcosn-2xdx277. xeax2dx=12aeax2+C278. 12e-x-222dx=121+erfx-2+C279. ex2dx=ex2j=0n-1a2jx2j+1+2n-1a2n-2ex2x2ndx，n0 其中a2j=1352j-12j+1=2j!j!22j+1+C280. -e-ax2dx=a the Gaussian integral281. 0x2ne-x2a2dx=2n!n!a22n+1282. lnaxdx=xlnax-x+C283. lnx2dx=xlnx2-2xlnx+2x+C284. lnaxndx=xlnaxn-nlnaxn-1dx285. dxlnx=lnlnx+lnx+i=2lnxiii!+C286. dxlnxn=-xn-1lnxn-1+1n-1dxlnxn-1+C，n1287. xmlnxdx=xm+1lnxm+1-1m+12+C，n-1288. xmlnxndx=xm+1lnxnm+1-nm+1xmlnxn-1dx+C，m-1289. lnxndxx=lnxn+1n+1+C，n-1290. lnxdxxm=-lnxm-1xm-1-1m-12xm-1，m1291. lnxndxxm=-lnxnm-1xm-1+nm-1lnxn-1dxxm，m1292. xmdxlnxn=-xm+1n-1lnxn-1+m+1n-1xmdxlnxn-1，n1293. xnlnaxmdx=xn+1lnaxmn+1-mn+1xnlnaxm-1dx，n-1294. lnaxmxdx=lnaxm+1m+1+C，m-1295. dxxlnax=lnlnax+C296. dxxnlnx=lnlnx+i=1-1in-1ilnxiii!+C297. dxxlnxn=-1n-1lnxn-1，n1298. sinlnxdx=x2sinlnx-coslnx+C299. coslnxdx=x2sinlnx+coslnx+C300. exxlnx-x-1xdx=exxlnx-x-lnx+C21