(一)含有axb的积分(a0) 1.
dx1=axbalnaxbC
2.(axb)dx=
1(axb)1C(1)
a(1)3.
x1=dxaxba2(axbblnaxb)C
x211dx=3(axb)22b(axb)b2lnaxbC 4.a2axb5.
1axbdx=x(axb)blnxC
6.
1aaxbdxlnC =22bxbxx(axb)7.
x1bdx=(lnaxb)C (axb)2a2axbx21b2dx=3(axb2blnaxb)C 8.2(axb)aaxb9.
11axbdxlnC =
xx(axb)2b(axb)b2(二)含有axb的积分
23(axb)C 3a2311.xaxbdx=(3ax2b)(axb)C 215a22(15a2x212abx8b2)(axb)3C 12.xaxbdx=3105a10.
axbdx=13.
x2dx=2(ax2b)axbC
3aaxb14.
x22dx=(3a2x24abx8b2)axbC 315aaxbdx15.=xaxb16.
1lnbaxbbC(b0)axbb
2axbarctanC(b0)bbx2dxaxbadx= bx2bxaxbaxb17.
axbdxdx=2axbb xxaxbaxbaxbadxdx= x2x2xaxb2218.
(三)含有xa的积分 19.
dx1x=arctanC x2a2aadxx2n3dx=(x2a2)n2(n1)a2(x2a2)n12(n1)a2(x2a2)n1 1xadxln=x2a22axaC
220.
21.
(四)含有axb(a0)的积分
1arctandxab22.2=axb1ln2ab23.
axCb(b0)
axbC(b0)axbx12dx=ax2b2alnaxbC
24.x2ax2bdx=xabadxax2b
.dxx(ax2b)=1x2252blnax2bC
26.
dx1ax2(ax2b)=bxdxbax2b
27.dxaax2b1x3(ax2b)=2b2lnx22bx2C 28.
dxx(ax2b)2=2b(ax2b)1dx2bax2b
(五)含有ax2bxc(a0)的积分
22ax2arctanb29.dx4acb4acb2Cax2bxc=12axbb2ln4acCb24ac2axbb24ac30.
x1bdxax2bxcdx=2alnax2bxc2aax2bxc(六)含有x2a2(a0)的积分 31.
dxx2a2=arshxaC1=ln(xx2a2)C 32.
dxx(x2a2)3=a2x2a2C
33.
xx2a2dx=x2a2C
34.
x1(x2a2)3dx=x2a2C
(b24ac)(b24ac)
35.
x2a22xaln(xx2a2)C dx=22x2a2x236.
x2(x2a2)3dxdx=xx2a2ln(xx2a2)C
1x2a2aC 37.=ln22axxxa38.
x2x2a2C =222axxadx39.
x2a22xaln(xx2a2)C xadx=2222x342232222(xa)dx=(2x5a)xaaln(xx2a2)C 8812241.xxadx=(x2a2)3C
340.42.x2xa42222xadx=(2xa)xaln(xx2a2)C
882243.
x2a2x2a2a22dx=xaalnC xxx2a2x2a2dx=ln(xx2a2)C 2xx44.
(七)含有x2a2(a0)的积分 45.
dxx2a2=
xxarchC1=lnxx2a2C xa=46.
dx(xa)xx2a2223xa2xa22C
47.
dx=x2a2C
48.
x(xa)223dx=1xa22C
49.
xx2a22xalnxx2a2C dx=22x2a2x250.
x2(x2a2)3dxx2a2dxdx=xx2a2lnxx2a2C
51.=1aarccosC ax52.
x2x2a2C =222axxa2253.
x2a22xalnxx2a2C xadx=22x34223222222(xa)dx=(2x5a)xaalnxxaC 8812255.xxadx=(x2a2)3C
354.56.x2xa42222xadx=(2xa)xalnxx2a2C
882257.
ax2a2dx=x2a2aarccosC
xxx2a2x2a222dxlnxxaC =2xx58.
(八)含有a2x2(a0)的积分 59.
dxa2x2=arcsinxC axC
60.
dx(ax)223=a2ax2261.
xa2x2dx=a2x2C
1ax2262.
x(ax)223dx=C
63.
x2a2x2axarcsinC dx=22aa2x2x264.
x2(a2x2)3dxdx=xa2x2arcsinxC a1aa2x2C 65.=ln22axxax66.
x2a2x2C =222axaxdx67.
x2a2x2axarcsinC axdx=22a22x34x2232222(ax)dx=(5a2x)axaarcsinC 88a12269.xaxdx=(a2x2)3C
368.70.x2xa4x2222axdx=(2xa)axarcsinC
88a2271.
a2x2aa2x222dx=axalnC xxa2x2a2x2xdxarcsinC =x2xa72.
(九)含有ax2bxc(a0)的积分 73.
dxax2bxc=1ln2axb2aax2bxcC a74.
ax2bxcdx=2axbax2bxc 4a
xax2bxc4acb28a3ln2axb2aax2bxcC
75.
dx=1ax2bxc ab2a3ln2axb2aax2bxcC
dxcbxax2276.
=12axbarcsinC
2ab4ac77.
2axbb24ac2axb2cbxaxdx=cbxaxarcsinC
324a8ab4acxcbxax2dx=1b2axbcbxax2arcsinC
32a2ab4ac78.
(十)含有xa或(xa)(bx)的积分 xbxb)C
79.
xaxadx=(xb)(ba)ln(xaxbxb80.
xaxaxadx=(xb)(ba)arcsinC bxbxbxxadxC=2arcsinbx(xa)(bx)81.
(ab)
82.
2xab(ba)2xa(xa)(bx)arcsinC (xa)(bx)dx=44bx (ab) (十一)含有三角函数的积分 83.sinxdx=cosxC
84.cosxdx=sinxC 85.tanxdx=lncosxC 86.cotxdx=lnsinxC 87.secxdx=lntan(x)C=lnsecxtanxC 4288.cscxdx=lntanxC=lncscxcotxC 289.secxdx=tanxC 90.cscxdx=cotxC 91.secxtanxdx=secxC 92.cscxcotxdx=cscxC
22x1sin2xC 24x1294.cosxdx=sin2xC
241n1n1nn295.sinxdx=sinxcosxsinxdx nn1n1nn1n296.cosxdx=cosxsinxcosxdx nndx1cosxn2dx97.= sinnxn1sinn1xn1sinn2xdx1sinxn2dx98.= nn1n2cosxn1cosxn1cosx1m1mnm2n99.cosxsinxdx=cosm1xsinn1xcosxsinxdx mnmn1n1cosm1xsinn1xcosmxsinn2xdx =mnmn93.sinxdx=
2100.sinaxcosbxdx=11cos(ab)xcos(ab)xC
2(ab)2(ab)101.sinaxsinbxdx=11sin(ab)xsin(ab)xC
2(ab)2(ab)102.cosaxcosbxdx=
11sin(ab)xsin(ab)xC
2(ab)2(ab)103.
2dx=absinxa2b2arctanatanxb2C22ab(a2b2)
x22bbadx12104.=lnC22xabsinxbaatanbb2a22atan105.
(a2b2)
2ababxdxarctan(tan)C=abcosxababab2(a2b2)
xdx1ab2106.=lnabcosxabbaxtan2tan107.
abbaCabba(a2b2)
dx1b=arctan(tanx)C a2cos2xb2sin2xaba1btanxadxln=a2cos2xb2sin2x2abbtanxaC
108.
11sinaxxcosaxC a2a12222110.xsinaxdx=xcosax2xsinax3cosaxC
aaa11111.xcosaxdx=2cosaxxsinaxC
aa12222112.xcosaxdx=xsinax2xcosax3sinaxC
aaa(十二)含有反三角函数的积分(其中a0)
xx22113.arcsindx=xarcsinaxC
aa109.xsinaxdx=
x2a2xx2xax2C 114.xarcsindx=()arcsin24a4ax3x12x222115.xarcsindx=arcsin(x2a)axC
3a9a2116.arccosdx=xarccosxaxa2x2C ax2a2xx2xax2C 117.xarccosdx=()arccos24a4ax3x12x222118.xarccosdx=arccos(x2a)axC
3a9a2xxa=dxxarctanln(a2x2)C aa2x12xa2120.xarctandx=(ax)arctanxC
a2a2119.arctanx3xa2a3xln(a2x2)C 121.xarctandx=arctanx3a66a2(十三)含有指数函数的积分
1xaC lna1axax123.edx=eC
a1axax124.xedx=2(ax1)eC
a1naxnn1axnax125.xedx=xexedx
aa122.adx=
x126.xadx=
nxxxx1aaxC 2lna(lna)1nxnn1xxaxadx lnalna1axeax(asinbxbcosbx)C 128.esinbxdx=22ab1axaxe(bsinbxacosbx)C 129.ecosbxdx=22ab127.xadx=
130.esinbxdx=
axn1eaxsinn1bx(asinbxnbcosbx) 222abnn(n1)b2axn2esinbxdx 2ab2n2131.ecosbxdx=
axn1axn1ecosbx(acosbxnbsinbx) 222abnn(n1)b2axn2ecosbxdx 222abn(十四)含有对数函数的积分 132.lnxdx=xlnxxC
dxxlnx=lnlnxC
1n11n134.xlnxdx=x(lnx)C
n1n1133.
135.(lnx)dx=x(lnx)n(lnx)136.x(lnx)dx=
nnn1dx
mn1nmn1xm1(lnx)nx(lnx)dx m1m1(十五)含有双曲函数的积分 137.shxdx=chxC 138.chxdx=shxC 139.thxdx=lnchxC
x1sh2xC 24x12141.chxdx=sh2xC
24140.shxdx=2(十六)定积分 142.143.
cosnxdx=sinnxdx=0
cosmxsinnxdx=0
144.
0,mncosmxcosnxdx= ,mn0,mn145.sinmxsinnxdx=
,mn146.
0sinmxsinnxdx=2000,mncosmxcosnxdx=
,mn2147. In= In=
sinxdx=cosnxdx
n20n1In2 nn1n3 Innn2n1n3Innn2
42,I1=1 (n为大于1的正奇数)
5331,I0= (n为正偶数)
4222一、 (系数不为0的情况)
a0xna1xn1limxbxmbxm101a0b0an0bmnmnmnm
lim1xe1x二、重要公式(1)
sinxlim1x0x
(2)x0 (3)
limna(ao)1n
n(4)nlimn1limarccotx0limexlimarctanx (5)
x2 (6)
xlimarctanxlimex02
(7)x (8)xlimarccotx
(9)x
(10)x (11)x0xlimx1
三、下列常用等价无穷小关系(
x0)
x arctanx
sinx
x tanxx e1xx arcsinxx a1xx
1cosx12x2
ln1x
xlna 1x1x
四、导数的四则运算法则
uvuv uvuvuv
五、基本导数公式
uuvuvv2 v1c0sinxcosxxx⑴ ⑵ ⑶
cosxsinxtanxsec2xcotxcsc2x⑷ ⑸ ⑹ secxsecxtanxcscxcscxcotx⑺ ⑻
⑼
eexx ⑽
aaxlnax1lnxx ⑾
⑿
logxa11arcsinx1x2xlna ⒀
arccosx ⒁
11x2
1arctanx1x2⒂
六、高阶导数的运算法则
1arccotx21x ⒃
x1⒄⒅
x12x
(1)
uxvxnnuxnvxn
(2)
cuxcunx
(3)
uaxbnnanunaxbkn (4)
uxvx
cuk0nnkxvx(k)
七、基本初等函数的n阶导数公式 (1)
xnnn! (2)neaxbnaneaxb (3)
axnaxlnna
(4)
sinaxbansinaxbn2 (5)
cosaxbnnancosaxbn2 n(6)
1axb1ann!axbn1lnaxb (7)
n1n1ann1!axbn
八、微分公式与微分运算法则 ⑴⑷⑺
dc0 ⑵
dxx1dx ⑶
dsinxcosxdx
dcosxsinxdx ⑸
dtanxsec2xdx ⑹
dcotxcsc2xdx
dsecxsecxtanxdxdexx ⑻
dcscxcscxcotxdxdlnx1dxx
⑼
edx ⑽daaxxxlnadx1 ⑾
1dlogadxxlna ⑿
darctanxdarcsinx⒀
1x2dx ⒁
darccosx11x2dx
⒂
11dxdarccotxdx221x1x ⒃
九、微分运算法则
⑴
duvdudv ⑵
dcucdu
⑶
duvvduudv
uvduudvd2vv⑷
十、基本积分公式
x1dxxdxclnxckdxkxc1⑴ ⑵ ⑶x
axxxadxcedxeccosxdxsinxclna⑷ ⑸ ⑹
x1dxsec2xdxtanxc2sinxdxcosxc⑺ ⑻cosx 112cscxdxcotxcdxarctanxc22⑼sinx ⑽1x
⑾
11x2dxarcsinxc
十一、下列常用凑微分公式 积分型 换元公式 faxbdx1faxbdaxba 1fxx1dxfxdxuaxb 1flnxdxflnxdlnxx xxxxfeedxfedefaxaxdx1faxdaxlna fsinxcosxdxfsinxdsinx2 ux ulnx uex fcosxsinxdxfcosxdcosx ftanxsecxdxftanxdtanx fcotxcscxdxfcotxdcotx 2uax usinx ucosx utanx ucotx
1dxfarctanxdarctanx21x 1farcsinxdxfarcsinxdarcsinx21x farctanx
uarctanx uarcsinx 十二、补充下面几个积分公式
tanxdxlncosxc cotxdxlnsinxc secxdxlnsecxtanxc cscxdxlncscxcotxc
11xdxarctanca2x2aa
11xadxlncx2a22axa
1a2x2dxarcsinxca
1x2a2dxlnxx2a2c
十三、分部积分法公式
⑴形如
naxxedx,令uxn,dveaxdx
形如
nxsinxdxnux令,dvsinxdx
形如
nxcosxdx令uxn,dvcosxdx
⑵形如
nxarctanxdx,令uarctanx,dvxndx
形如
nxlnxdx,令ulnx,dvxndx
axesinxdxeaxcosxdxueax,sinx,cosx⑶形如,令均可。
十四、第二换元积分法中的三角换元公式 (1)a2x2
2222xasint (2) ax xatant (3)xa xasect
【特殊角的三角函数值】
(1)sin00 (2)
sin631sinsin132 (4)2 (3)2) (5)sin0
31coscos02 (3)32 (4)2) (5)cos1 3tan3tan3 (3)32不存在 (5)tan0 (4)
cot (3)
(1)cos01 (2)
cos6(1)tan00 (2)
tan6(1)cot0不存在 (2)
十五、三角函数公式
1.两角和公式
cot63333cot(4)
20(5)cot不存在
sin(AB)sinAcosBcosAsinB sin(AB)sinAcosBcosAsinB cos(AB)cosAcosBsinAsinB cos(AB)cosAcosBsinAsinB
tanAtanBtanAtanBtan(AB)1tanAtanB 1tanAtanB cotAcotB1cotAcotB1cot(AB)cot(AB)cotBcotA cotBcotA tan(AB)
2.二倍角公式
sin2A2sinAcosA cos2Acos2Asin2A12sin2A2cos2A1
tan2A
3.半角公式
2tanA1tan2A
sinA1cosAA1cosAcos2222
tan
A1cosAsinAA1cosAsinAcot21cosA1cosA 21cosA1cosA
4.和差化积公式
sinasinb2sinababababcossinasinb2cossin22 22 ababababcosacosb2coscoscosacosb2sinsin22 22
tanatanb
5.积化和差公式
sinabcosacosb
11sinasinbcosabcosabcosacosbcosabcosab22
11sinacosbsinabsinabcosasinbsinabsinab22
6.万能公式
a1tan22sinacosaa1tan21tan22
2tan
7.平方关系
a2a2
a2tanaa1tan22
2tansin2xcos2x1 sec2xtan2x1 csc2xcot2x1
8.倒数关系
tanxcotx1 secxcosx1 cscxsinx1
9.商数关系
tanx
sinxcosxcotxcosx sinx
十六、几种常见的微分方程
dyfxgyfxg1ydxf2xg2ydy01.可分离变量的微分方程:dx , 1
dyyfdxx 2.齐次微分方程:
dypxyQxdx3.一阶线性非齐次微分方程: 解为:
ye
pxdxQxepxdxdxc
因篇幅问题不能全部显示,请点此查看更多更全内容