(完整版)常用函数积分表(增强版).doc
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(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=1aarcsinh2ax+b4ac-b2+C,a>0,4ac-b2>0
117.dxax2+bx+c=1aln2ax+b+C,a>0,4ac-b2=0
118.dxax2+bx+c=-1-aarcsin2ax+bb2-4ac+C,a<0,4ac-b2<0
119.dxax2+bx+c3=4ax+2b4ac-b2ax2+bx+c+C
120.dxax2+bx+c5=4ax+2b34ac-b2ax2+bx+c1ax2+bx+c+8a4ac-b2+C
121.dxax2+bx+c2n+1=4ax+2b2n-14ac-b2ax2+bx+c2n-1+8an-12n-14ac-b2dxax2+bx+c2n-1+C循环计算
122.xdxax2+bx+c=ax2+bx+ca-b2adxax2+bx+c+C
123.xdxax2+bx+c3=-2bx+4c4ac-b2ax2+bx+c+C
124.xdxax2+bx+c2n+1=-12n-1aax2+bx+c2n-1-b2adxax2+bx+c2n-1+C
125.dxxax2+bx+c=-1cln2acx2+bcx+c2+bx+2cx+C
126.dxxax2+bx+c=-1carcsinhbx+2cx4ac-b2+C
127.sin2xdx=x2-sin2x4+C
128.1-sinxdx=cvsxdx=2cosx2+sinx2cosx2-sinx2,cvsx=21+sinx,其中cvsx是conversine函数
129.sinnaxdx=-sinn-1axcosaxan+n-1nsinn-2axdx+C循环计算
130.sinaxxdx=i=0∞-1iax2i+12i+12i+1!
+C
131.sinaxxndx=-sinaxn-1xn-1+an-1cosaxxn-1dx
132.cosnaxdx=1ancosn-1axsinax+n-1ncosn-2axdx+C,n≥2
133.cos2xdx=x2+sin2x4+C
134.cosaxxdx=lnax+i-1∞-1iax2i2i2i!
,n≠1
135.cosaxxndx=-cosaxn-1xn-1-an-1sinaxxn-1dx,n≠1
136.sinaxcosaxdx=12asin2ax
137.sinaxsinbxdx=sina-bx2a-b-sina+bx2a+b+C,a2≠b2
138.sinaxcosbxdx=-cosa+bx2a+b-cosa-bx2a-b+C,a2≠b2
139.cosaxcosbxdx=sina-bx2a-b+sina+bx2a+b+C,a2≠b2
140.sinaxcosaxdx=-cos2ax4a+C,a≠0
141.sinnaxcosaxdx=sinn+1axn+1a+C,a≠0,n≠-1
142.cosnaxsinaxdx=-cosn+1axn+1a+C,a≠0,n≠-1
143.tanaxdx=sinaxcosaxdx=-1alncosax+C,a≠0
144.cotaxdx=cosaxsinaxdx=1alnsinax+C,a≠0
145.sinnaxcosmaxdx=-sinn-1axcosm+1axam+n+n-1m+nsinn-2axcosmaxdx+C=sinn+1axcosm-1axam+n+m-1n+msinnaxcosm-2axdx+C,a≠0,m+n≠0循环计算
146.sinaxsinbxdx=xsina-b2a-b-xsina+b2a+b+C,a≠b
147.dxsinaxcosax=1alntanax+C
148.dxsinaxcosnax=1an-1cosn-1ax+dxsinaxcosn-2ax,n≠1
149.dxcosaxsinnax=-1an-1sinn-1ax+dxcosaxsinn-2ax,n≠1
150.sinaxdxcosnax=1an-1cosn-1ax+C,n≠1
151.sin2axdxcosax=-1asinax+1alntanπ4+ax2+C
152.sin2axdxcosnax=sinaxan-1cosn-1ax-1n-1dxcosn-2ax,n≠1
153.sinnaxdxcosax=-sinn-1axan-1+sinn-2axdxcosax+C
154.sinnaxdxcosmax=sinn+1axam-1cosm-1ax-n-m+2m-1sinnaxdxcosm-2ax+C=-sinn-1axan-mcosm-1ax+n-1n-msinn-2axdxcosmax+C=sinn-1axam-1cosm-1ax-n-1m-1sinn-1axdxcosm-2ax+C,m≠1,m≠n
155.cosaxdxsinnax=-1an-1sinn-1ax+C
156.cos2axdxsinax=1acosax+lntanax2+C
157.cos2axdxsinnax=-1n-1cosaxasinn-1ax+dxsinn-2ax+C,n≠1
158.cosnaxdxsinmax=-cosn+1axam-1sinm-1ax-n-m-2m-1cosnaxdxsinm-2ax+C=cosn-1axan-msinm-1ax+n-1n-mcosn-2axdxsinmax+C=-cosn-1axam-1sinm-1ax-n-1m-1cosn-2axdxsinm-2ax+C,m≠1,m≠n
159.dxb+csinax=-2ab2-c2arctanb-cb+ctanπ4-ax2+C,a≠0,b2>c2
160.dxb+csinax=-1ac2-b2lnc+bsinax+c2-b2cosaxb+csinax+C,a≠0,b2 161.dx1+sinax=-1atanπ4-ax2+C,a≠0 162.dx1-sinax=1atanπ4+ax2+C,a≠0 163.xdx1+sinax=xatanax2-π4+2c2lncosax2-π4+C 164.xdx1-sinax=xacotπ4-ax2+2c2lnsinπ4-ax2+C 165.sinaxdx1±sinax=±x+1ctanπ4∓ax2+C 166.dxb+ccosax=2ab2-c2arctanb-cb+ctanax2+C,a≠0,b2>c2 167.dxb+ccosax=1ac2-b2lnc+bcosax+c2-b2sinaxb+ccosax+C,a≠0,b2 168.dx1+cosax=1atanax2+C,a≠0 169.dx1-cosax=-1acotax2+C,a≠0 170.xdx1+cosax=xatanax2+2a2lncosax2+C,a≠0 171.xdx1-cosax=-xacotax2+2a2lnsinax2+C,a≠0 172.cosaxdx1+cosax=x-1atanax2+C 173.cosaxdx1-cosax=-x-1acotax2+C 174.cosaxcosbxdx=xsina-b2a-b+xsina+b2a+b+C,a≠b 175.dxcosax±sinax=12alntanax2±π8+C 176.dxcosax+sinax2=12atanax∓π4+C 177.dxcosx+sinxn=1n-1sinx-cosxcosx+sinxn-1-2n-2dxcosx+sinxn-2+C 178.dxcosax+sinaxn= 179.cosaxdxcosax+sinax=x2+12alnsinax+cosax+C 180.cosaxdxcosax-sinax=x2-12alnsinax-cosax+C 181.sinaxdxcosax+sinax=x2-12alnsinax+cosax+C 182.sinaxdxcosax-sinax=x2-12alnsinax-cosax+C 183.cosaxdxsinax1+cosax=-14atan2ax2+12alntanax2+C 184.cosaxdxsinax1-cosax=-14acot2ax2-12alntanax2+C 185.sinaxdxcosax1+sinax=14acot2ax2+π4+12alntanax2+π4+C 186.sinaxdxcosax1-sinax=14atan2ax2+π4-12alntanax2+π4+C 187.sinaxtanaxdx=1alnsecax+tanax-sinax+C 188.tannaxdxsin2ax=1an-1tann-1ax,n≠1 189.tannaxdxcos2ax=1an+1tann+1ax,n≠-1 190.cotnaxdxsin2ax=1an+1cotn+1ax,n≠-1 191.cotnaxdxcos2ax=1a1-ntan1-nax,n≠1 192.tanmaxcotnax=1am+n-1tanm+n-1ax-tanm-2axcotnaxdx,m+n≠1 193.xsinaxdx=1a2sinax-xacosax+C,a≠0 194.xcosaxdx=cosaxa2+xsinaxa+C 195.xnsinaxdx=-xnacosax+naxn-1cosaxdx,a≠0循环计算 196.xncosaxdx=xnasinax-naxn-1sinaxdx,a≠0循环计算 197.tanaxdx=-1alncosax+C,a≠0 198.cotaxdx=1alnsinax+C,a≠0 199.tan2axdx=1atanax-x+C,a≠0 200.cot2axdx=-1acotax-x+C,a≠0 201.tannaxdx=tann-1axan-1-tann-2axdx,a≠0,n≠1循环计算 202.cotnaxdx=-cotn-1axan-1-cotn-2axdx,a≠0,n≠1循环计算 203.dxtanax+1=x2+12alnsinax+cosax+C 204.dxtanax-1=-x2+12alnsinax-cosax+C 205.tanaxdxtanax+1=x2-12alnsinax+cosax+C 206.tanaxdxtanax-1=x2+12alnsinax-cosax+C 207.dx1+cotax=tanaxdxtanax+1=x2-12alnsinax+cosax+C 208.dx1-cotax=tanaxdxtanax-1=x2+12alnsinax-cosax+C 209.secaxdx=1alnsecax+tanax+C=1alntanax2+π4,a≠0 210.cscaxdx=-1alncscax+cotax+C=1alntanax2+C,a≠0 211.secnaxdx=secn-2axtanaxan-1+n-2n-1secn-2axdx,a≠0,n≠1循环计算 212.cscnaxdx=-cscn-2axcotaxan-1+n-2n-1cscn-2axdx,a≠0,n≠1循环计算 213.secnaxtanaxdx=secnaxna+C,a≠0,n≠0 214.cscnaxcotaxdx=-cscnaxna+C,a≠0,n≠0 215.dxsecx
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