(一)含有axb的积分(a0)
dx1
1.=lnaxbC
axba
2.(axb)dx=
1
(axb)1C(1)
a(1)
x1
3.dx=(axbblnaxb)C
axba2
x2114.dx=(axb)22b(axb)b2lnaxbC
a32axbdx1axb
5.=lnC
bxx(axb)
6.7.
dx1aaxb=lnC
bxb2xx2(axb)
1bx
=dx(lnaxb)C
(axb)2a2axb
x21b2
8.dx=(axb2blnaxb)C
(axb)2a3axb
9.
dx11axb
=lnCb(axb)b2xx(axb)2
(二)含有axb的积分
2
10.axbdx=(axb)3C
3a2
11.xaxbdx=(3ax2b)(axb)3C
15a2
2
12.x2axbdx=(15a2x212abx8b2)(axb)3C
105a3
13.14.
2x
dx=(ax2b)axbC
3a2axb
x22=dx(3a2x24abx8b2)axbC15a3axb
dx
15.=
xaxb
1lnbaxbb
C(b0)
axbb
2axbarctanC(b0)
bb
16.17.18.
x2
dxaxbadx
=
bx2bxaxbaxb
axbdx
dx=2axbbxxaxbaxbaxbadx
dx=x2x2xaxb
(三)含有x2a2的积分19.
dx1x=arctanC
x2a2aa
dxx2n3dx
20.=
(x2a2)n2(n1)a2(x2a2)n12(n1)a2(x2a2)n1
21.
dx1xa
=lnCx2a22axa
(四)含有ax2b(a0)的积分
1
arctan
dxab
22.=
ax2b1
ln
2ab
a
xCb
(b0)
axb
C(b0)
axb
x1
23.dx=lnax2bC
ax2b2a
x2xbdx24.dx=
ax2baaax2b
1x2dx
25.=lnC
x(ax2b)2bax2b1adxdx
26.=
x2(ax2b)bxbax2b
27.dxx=alnax2bx1
C
3(ax2b)2b222bx2
28.
dxx1dx
(axb)=
222b(ax2b)2bax2b
(五)含有ax2bxc(a0)的积分
22ax29.dx
arctanb4acb2
4acbC2
axc=2bx
12axbb24ac
blnbbC
24ac2ax24ac30.xaxdx=1lnax2bxc
b2bxc2a2adx
ax2bxc
(六)含有x2a2(a0)的积分31.dx
=arshx
x2a2aC=ln(xx21a2)C
32.dx=
x(xC
2a2)3
a2x2a2
33.xxdx=x2a2C
2a2
34.
x(xdx=
12a2)3
xC
2a2
35.
x2
xxa2a2
2ln(xxdx=2a2
22x2a2)C36.
x2(x2adx=
xln(xx2a2)C
2)3
x2a2
37.dx
=1x2a2a
xx2a2
alnxC
38.
dx
=x2a2x2
x2a2
aC
2x39.
x2a2dx=xa2
2x2a22
ln(xx2a2)C(b24ac)(b24ac)
x3
40.(x2a2)3dx=(2x25a2)x2a2a4ln(xx2a2)C
881
41.xx2a2dx=(x2a2)3C
3
42.x243.
xa4
x2a2dx=(2x2a2)x2a2ln(xx2a2)C
88
x2a2x2a2a
dx=x2a2alnCxxx2a2x2a2
dx=ln(xx2a2)Cx2x
44.
(七)含有x2a2(a0)的积分45.46.47.48.
xx
=archC=lnxx2a2C
1xax2a2
dx
dx(x2a2)3
xx2a2
=
xa2x2a2
C
dx=x2a2C
x(x2a2)3
dx=
1x2a2
C
49.
xa2
x2a2lnxx2a2Cdx=22x2a2
x2
50.
x2(x2a2)3
dx
dx=
xx2a2
lnxx2a2C
1a
51.=arccosC
axxx2a2
52.
x2
x2a2=C
a2xx2a2
dx
53..
xa2
x2a2lnxx2a2Cx2a2dx=22
x3
(x2a2)3dx=(2x25a2)x2a2a4lnxx2a2C
88
1
55.xx2a2dx=(x2a2)3C
3
56.x257.
xa4
x2a2dx=(2x2a2)x2a2lnxx2a2C
88
ax2a2
22=xaaarccosCdx
xx
x2a2x2a2
dx=lnxx2a2Cx2x
58.
(八)含有a2x2(a0)的积分59.60.61.62.
x
=arcsinC
aa2x2dx
dx(a2x2)3
xa2x2
=
xa2a2x2
C
dx=a2x2C
x(a2x2)3
dx=
1a2x2
C
63.
xa2x22axarcsinCdx=22aa2x2
x
arcsinCdx=
aa2x2(a2x2)3
dx
x2
.
x2x
1aa2x2
65.=lnC
axxa2x2
66.
x2
a2x2=C
a2xa2x2
dx
67.
xa2xa2x2arcsinCa2x2dx=22a
x3x
68.(a2x2)3dx=(5a22x2)a2x2a4arcsinC
88a1
69.xa2x2dx=(a2x2)3C
3
70.x271.72.
xa4x
a2x2dx=(2x2a2)a2x2arcsinC
88a
a2x2aa2x2
dx=a2x2alnCxxa2x2a2x2x
dx=arcsinCx2xa
(九)含有ax2bxc(a0)的积分73.74.75.76.77.78.
1
=ln2axb2aax2bxcC
aax2bxc
dx
2axb
ax2bxcdx=ax2bxc
4a
1dx=ax2bxc
aax2bxc
xdxcbxax2
=
12axbarcsinCab24ac
2axbb24ac2axb
22=cbxaxdxcbxaxarcsinC
4a8a3b24ac
1b2axb
dx=cbxax2arcsinC
acbxax22a3b24ac
x
xa
(十)含有或(xa)(bx)的积分
xb
79.80.81.
xaxa
dx=(xb)(ba)ln(xbxb
xaxb)C
xaxa
=dx(xb)(ba)arcsin
bxbxxa
Cbx
dx
=2arcsin
(xa)(bx)
xa
Cbx
(ab)
82.
2xab(ba)2xa
(xa)(bx)arcsinC(xa)(bx)dx=
44bx
(十一)含有三角函数的积分
83.sinxdx=cosxC84.cosxdx=sinxC85.tanxdx=lncosxC86.cotxdx=lnsinxC
87.secxdx=lntan()C=lnsecxtanxC
x
88.cscxdx=lntanC=lncscxcotxC
2
4
x2
.sec2xdx=tanxC90.csc2xdx=cotxC91.secxtanxdx=secxC92.cscxcotxdx=cscxC
x1
93.sin2xdx=sin2xC
24x1
94.cos2xdx=sin2xC
241n1
sinn2xdx95.sinnxdx=sinn1xcosx
nn1n1
cosn2xdx96.cosnxdx=cosn1xsinx
nn
dx1cosxn2dx
97.=
sinnxn1sinn1xn1sinn2xdx1sinxn2dx
98.=
cosnxn1cosn1xn1cosn2x
1m1
m1n1cosm2xsinnxdx99.cosmxsinnxdx=cosxsinx
mnmn
1n1
cosmxsinn2xdx=cosm1xsinn1x
mnmn
100.sinaxcosbxdx=
11
cos(ab)xcos(ab)xC
2(ab)2(ab)
11
101.sinaxsinbxdx=sin(ab)xsin(ab)xC
2(ab)2(ab)
11
102.cosaxcosbxdx=sin(ab)xsin(ab)xC
2(ab)2(ab)
x
atanb2dx2103.=arctanC
absinxa2b2a2b2
(a2b2)
x
2a2atanbbdx12104.=lnC
absinxb2a2atanxbb2a2
2
2ababxdx
105.=arctan(tan)C
ababab2abcosx
(a2b2)
(a2b2)
x
tan
dx1ab2106.=lnabcosxabbax
tan2abbaCabba
(a2b2)
dx1b
107.=arctan(tanx)C
a2cos2xb2sin2xaba
108.
dx1btanxa
=lnC2abbtanxaa2cos2xb2sin2x
11
109.xsinaxdx=sinaxxcosaxC
a2a122
110.x2sinaxdx=x2cosaxxsinaxcosaxC
aa2a311
111.xcosaxdx=cosaxxsinaxC
a2a122
112.x2cosaxdx=x2sinaxxcosaxsinaxC
aa2a3
(十二)含有反三角函数的积分(其中a0)113.arcsindx=xarcsina2x2C
x2a2xxx
114.xarcsindx=()arcsina2x2C
24a4ax3x1x
115.x2arcsindx=arcsin(x22a2)a2x2C
3a9a
x
axa
xx
116.arccosdx=xarccosa2x2C
aa
x2a2xxx
117.xarccosdx=()arccosa2x2C
24a4ax3x1x
118.x2arccosdx=arccos(x22a2)a2x2C
3a9a
xxa
119.arctandx=xarctanln(a2x2)C
aa2x1xa
120.xarctandx=(a2x2)arctanxC
a2a2
x3xaa3x
121.x2arctandx=arctanx2ln(a2x2)C
3a66a
(十三)含有指数函数的积分
1
122.axdx=axC
lna1
123.eaxdx=eaxC
a1
124.xeaxdx=(ax1)eaxC
a21n
125.xneaxdx=xneaxxn1eaxdx
aa
x1
126.xaxdx=axaxC
lna(lna)2
1n
nxxn1axdxxalnalna
1
128.eaxsinbxdx=eax(asinbxbcosbx)C
a2b2
1
129.eaxcosbxdx=eax(bsinbxacosbx)C
a2b2
1
130.eaxsinnbxdx=eaxsinn1bx(asinbxnbcosbx)
a2b2n2
1
131.eaxcosnbxdx=eaxcosn1bx(acosbxnbsinbx)
a2b2n2
127.xnaxdx=
(十四)含有对数函数的积分132.lnxdx=xlnxxC
dx
133.=lnlnxC
xlnx
11
134.xnlnxdx=xn1(lnx)C
n1n1
135.(lnx)ndx=x(lnx)nn(lnx)n1dx
136.xm(lnx)ndx=1n
m1xm1(lnx)n
m1
xm(lnx)n1dx(十五)含有双曲函数的积分137.shxdx=chxC138.chxdx=shxC139.thxdx=lnchxC140.sh2xdx=x12
4
sh2xC
141.ch2xdx=x21
4
sh2xC
(十六)定积分
142.cosnxdx=sinnxdx=0
143.cosmxsinnxdx=0
144.
cosmxcosnxdx=
0,mn
,mn
145.
sinmxsinnxdx=
0,mn
,mn
0,mn
146.sinmxsinnxdx=cosmxcosnxdx=
002
,mn147.I=
sin2nxdx=2n
cosnxdx
n01
0
I=
nnI
n2
In1n3n42
nn2L5
3
(n为大于1的正奇数),Inn1nn3n2L3142
2(n为正偶数),I=
02
I=11
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