(完整版)常用函数积分表(增强版).doc

(完整版)常用函数积分表(增强版).doc

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1.fx+gxdx=fxdx+gxdx

2.fx-gxdx=fxdx-gxdx

3.fxdgx=fxgx-gxdfx

4.axdx=axlna+C，a≠1，a>0

5.xndx=xn+1n+1+C，n≠-1

6.1xdx=lnx+C

7.exdx=ex+C

8.sinxdx=-cosx+C

9.cosxdx=sinx+C

10.sec2xdx=tanx+C

11.csc2xdx=-cotx+C

12.secxtanxdx=secx+C

13.cscxcotxdx=-cscx+C

14.ax+bndx=ax+bn+1an+1+C，a≠0，n≠-1

15.dxax+b=1alnax+b+C，a≠0

16.xax+bndx=ax+bn+1a2ax+bn+2-bn+1+C，a≠0，n≠-1,-2

17.xax+bdx=xa-ba2lnax+b+C，a≠0

18.xax+b2dx=1a2lnax+b+bax+b+C，a≠0

19.x2ax+bdx=12a3ax+b2-4bax+b+2b2lnax+b+C

20.x2ax+b2dx=1a3ax+b-2blnax+b-b2ax+b+C

21.x2ax+b3dx=1a3lnax+b+2bax+b-b22ax+b2+C

22.x2ax+bndx=1a3-1n-3ax+bn-3+2bn-2ax+bn-2-b2n-1ax+bn-1+C，n≠1,2,3

23.dxxax+b=1blnxax+b+C，b≠0

24.dxx2ax+b=-1bx+ab2lnax+bx+C

25.dxx2ax+b2=-a1b2ax+b+1ab2x-2b3lnax+bx+C

26.xax+bdx=215a23ax-2bax+b32+C

27.x2ax+bdx=2105a315a2x2-12abx+8b2ax+b32+C

28.ax+bndx=2ax+bn+2an+2+C，a≠0，n≠-2

29.xnax+bdx=2a2n+3xnax+b32-2nba2n+3xn-1ax+bdx循环计算

30.ax+bxdx=2ax+b+bdxxax+b=2ax+b-2barctanhax+bb+C

31.dxxax+b=2-barctanax+b-b+C，b<0

32.dxxax+b=1blnax+b-bax+b+b+C，b>0

33.ax+bx2dx=-ax+bx+a2dxxax+b+C

34.ax+bxndx=-ax+b32bn-1xn-1-2n-5a2bn-1ax+bxn-1dx，n≠1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，b≠0

37.dxxnax+b=-ax+bbn-1xn-1-2n-3a2bn-1ax+bxn-1dx，n≠1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，a≠0

40.dxa2+x22=x2a2a2+x2+12a3arctanxa+C，a≠0

41.dxa2-x2=12alna+xa-x+C=1aarctanhxa+C，a≠0，a>x

42.dxa2-x22=x2a2a2-x2+14a3lnx+ax-a+C

43.1x2-a2dx=12alnx-ax+a+C=-1aarccothxa+C，a≠0，x>a

44.dxa2+x2=lnx+a2+x2+C

45.a2+x2dx=x2a2+x2+a22lnx+a2+x2+C

46.a2+x23dx=xa2+x234+38a2xa2+x2+38a4lnx+a2+x2+C

47.a2+x25dx=xa2+x256+524a2xa2+x23+516a4xa2+x2+516a6lnx+a2+x2+C

48.xa2+x22n+1dx=a2+x22n+32n+3+C

49.x2a2+x2dx=x8a2+2x2a2+x2-a48lnx+a2+x2+C

50.x2a2+x23dx=xa2+x256-a2xa2+x224-a4xa2+x216-a616lnx+a2+x2+C

51.x3a2+x2dx=a2+x255-a2a2+x233+C

52.x3a2+x23dx=a2+x277-a2a2+x255+C

53.x3a2+x22n+1dx=a2+x22n+52n+5-a2a2+x22n+32n+3+C

54.x4a2+x2dx=x3a2+x236-a2xa2+x238+a4xa2+x216+a616lnx+a2+x2+C

55.x4a2+x23dx=x3a2+x258-a2xa2+x2516+a4xa2+x2364+3a6xa2+x2128+3a8128lnx+a2+x2+C

56.x5a2+x2dx=a2+x277-2a2a2+x255+a4a2+x233+C

57.x5a2+x23dx=a2+x299-2a2a2+x277+a4a2+x255+C

58.x5a2+x22n+1dx=a2+x22n+72n+7-2a2a2+x22n+52n+5+a4a2+x22n+32n+3+C

59.a2+x2xdx=a2+x2-alna+a2+x2x+C=a2+x2-aarcsinhax+C

60.a2+x23xdx=a2+x233+a2a2+x2-a3lna+a2+x2x+C

61.a2+x25xdx=a2+x255+a2a2+x233+a4a2+x2-a5lna+a2+x2x+C

62.a2+x277dx=a2+x277+a2a2+x255+a4a2+x233+a6a2+x2-a7lna+a2+x2x+C

63.a2+x2x2dx=lnx+a2+x2-a2+x2x+C

64.x2a2+x2dx=-a22lnx+a2+x2+xa2+x22+C=-a22arcsinhxa+xa2+x22+C

65.dxxa2+x2=-1alna+a2+x2x+C=-1aarcsinhax+C

66.dxx2a2+x2=-a2+x2a2x+C，a≠0

67.dxa2-x2=arcsinxa+C，a≠0，x≤a

68.a2-x2dx=x2a2-x2+a22arcsinxa+C，a≠0，x≤a

69.a2-x2dx=12xa2-x2-sgnxarccoshxa+C，x≥a

70.xa2-x2dx=-a2-x233+C，x≤a

71.x2a2-x2dx=a48arcsinxa-18xa2-x2a2-2x2+C，a≠0

72.a2-x2xdx=a2-x2-alna+a2-x2x+C，x≤a

73.a2-x2x2dx=-arcsinxa-a2-x2x+C，a≠0

74.x2a2-x2dx=a22arcsinxa-xa2-x22+C，a≠0，xa2-x2

75.dxxa2-x2=-1alna+a2-x2x+C，a≠0

76.dxx2a2-x2=-a2-x2a2x+C，a≠0

77.dxx2-a2=lnx+x2-a2+C

78.x2-a2dx=x2x2-a2-a22lnx+x2-a2+C

79.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，n≠2循环计算

81.xx2-a2ndx=x2-a2n+2n+2+C，n≠-2

82.x2x2-a2dx=x82x2-a2x2-a2-a48lnx+x2-a2+C

83.x2-a2xdx=x2-a2-aarcsecxa+C=x2-a2-aarccosax+C，a≠0

84.xdxx2-a2=x2-a2+C

85.xdxx2-a23=-1x2-a2+C

86.xdxx2-a25=-13x2-a23+C

87.xdxx2-a27=-15x2-a25+C

88.xdxx2-a22n+1=-12n-1x2-a22n-1+C

89.x2-a2x2dx=lnx+x2-a2-x2-a2x+C

90.x2x2-a2dx=a22lnx+x2-a2+x2x2-a2+C

91.x2x2-a23dx=-xx2-a2+lnx+x2-a2a+C

92.x4x2-a2dx=x3x2-a24+38a2xx2-a2+38a4lnx+x2-a2a+C

93.x4x2-a23dx=xx2-a22-a2xx2-a2+32a2lnx+x2-a2a+C

94.x4x2-a25dx=-xx2-a2-x33x2-a23+lnx+x2-a2a+C

95.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>m≥0

96.dxx2-a23=-xa2x2-a2+C

97.dxx2-a25=1a4xx2-a2-x33x2-a23+C

98.dxx2-a27=-1a6xx2-a2-2x33x2-a23+x55x2-a25+C

99.dxx2-a29=1a8xx2-a2-2x33x2-a23+3x55x2-a25-x77x2-a27+C

100.x2x2-a25dx=-x33a2x2-a23+C

101.x2x2-a27dx=1a4x33x2-a23-x55x2-a25+C

102.x2x2-a29dx=-1a6x33x2-a23-2x55x2-a25+x77x2-a27+C

103.dxxx2-a2=1aarcsecxa+C，a≠0

104.dxx2x2-a2=x2-a2a2x+C，a≠0

105.dxax2+bx+c=24ac-b2arctan2ax+b4ac-b2，4ac-b2>0

106.dxax2+bx+c=2b2-4acarctanh2ax+bb2-4ac=1b2-4acln2ax+b-b2-4ac2ax+b+b2-4ac，4ac-b2<0

107.dxax2+bx+c=-22ax+b，4ac-b2=0

108.dxax2+bx+c=12alnax2+bx+c-b2adxax2+bx+c+C

109.mx+nax2+bx+cdx=m2alnax2+bx+c+2an-bma4ac-b2arctan2ax+b4ac-b2+C，4ac-b2>0

110.mx+nax2+bx+cdx=m2alnax2+bx+c+2an-bmab2-4acarctanh2ax+bb2-4ac+C，4ac-b2<0

111.mx+nax2+bx+cdx=m2alnax2+bx+c-2an-bma2ax+b+C，4ac-b2=0

112.dxax2+bx+cn=2ax+bn-14ac-b2ax2+bx+cn-1+2n-32an-14ac-b2dxax2+bx+cn-1+C

113.xax2+bx+cndx=bx+2cn-14ac-b2ax2+bx+cn-1-b2n-3n-14ac-b2dxax2+bx+cn-1+C

114.dxxax2+bx+c=12clnx2ax2+bx+c-b2cdxax2+bx+c+C

115.dxax2+bx+c=1aln2a2x2+abx+ac+2ax+b+C，a>0

116.dxax2+bx+c