2021年高等數(shù)學(xué)積分公式大全

**歐陽光明*創(chuàng)編2021.03.07**歐陽光明*創(chuàng)編2021.03.07常用積分公式歐陽光明（2021.03.07）（一）含有axb的積分a0)1．dx

＝1lnaxbCaxb a2．(axb)dx＝

1 (axb)1C（1）a(1)3．

dx

1(axbblnaxb)Caxb a24．

x2 dx＝ (axb)2b(axb)blnaxb C11 11 axbxaxb a3axbx5．

dxx(axb)

＝1ln b6． dxx2(axx 1dx＝x 1dx＝(lnax(axb)2 a2bbaxb)Cx2 1dx＝(ax(axb)2 a3b2blnaxbb2axb

＝1aln bx b2axbx8．a(chǎn)xbx

)C9．

dx ＝ 1

1ln Cx(axb)2axaxb

b(axb) b2的積分axbx23a(axb)3axbx23a(axb)3x axbdx

2 (3ax2b) C(axb)(axb)3(axb)312．x2

axbdx

2axbaxb

(15a2x212abx8b2) C13．

xax

dx

2axbaxb

(ax2b) C14．

x2ax

dx

215a3

(3a2x24abx8b2) Cdx

1ln

axb axaxb axb x axbbaxbx axbbaxbbbb

2 arctan C (b0)x2 axx2 axb

＝ axba dxx axb17x axb

axbdx＝x

bx 2baxbx axaxbx axb18．

axbdx＝ axba dxx axbx2 x 2x axb（三）含有x2a2的積分19．

dxx2a2

=1a

xaC20．

dx = x 2n3 dx(x2a2)n 2(n1)a2(x2a2)n1 2(n1)a2

(x2a2)n121．

dx

1ln Cxaxax2xaxa（四）含有ax2b(a0)的積分 dx

1abab abab＝

xC (b0)ax baxax bax b

22ab

C (b0) xax2

dx＝12a

lnax2bC24． x2ax2

dx＝xba a

dxax2b dxx(ax2

＝1ln x2 Cax2ax2b dx

＝1a dxx2(ax2b)

bx b

ax2b dxx3(ax2

＝a lnax2b2bax2b

1 C2bx2 dx(ax2b)2

＝ x2b(ax2

12b

dxax2（五）含有ax2bxc(a0)的積分29 ． dx ＝ax2bxc4acb2 4acb24ac4acb22ax2axb b24ac2axb b24ac1

C (b24ac)b2b24acx

C (b24ac)＝1 b dx ax2bxc

lnax2bxc 2a 2a

ax2bxcx2x2a2

(a0)的積分x2ax2a2

＝arshxa

C＝ln(x x2a2)C132．33．

dx ＝ x C(x2a2)3a2 x(x2a2)3a2 x2a2x2a2x2a234．

x(x2a2)3

dx＝ 1 Cx2ax2a2

x2 dx

a2x2a2x x2a2 ln(x xx2a2x x2a2x2x2a2(x2(x2a2)3

x2 dx＝ x

ln(x x2a2)C37．38．39．

dx ＝1x x2x x2a2x2 x2ax2 x2a2x x2a2x2ax x2a2

Cx2a2axx2x2a2axa2 ln(x x2a2)C2 2(x(x2a2)3dxx2ax2a2(2x25a2) a4ln(x x2a2)C8 813(x2a2)341．x x2a13(x2a2)342．x2

x2a2dx

xx2a2(2x2x2a2

a4 ln(x x2a2)C8 843．44．

x2a2dxxx2a2dxx2

＝ aln Cx2a2x2a2axx2a2＝ x2a2x2a2axx2a2x2x2a2

(a0)的積分x2ax2a2

xxxarchxx

C1

lnx Cx2a2x2a247．48．49．

dx ＝(x2a2)3(x2a2)3a2 x2a2x2a2x2a2(x2a2(x2a2)3x x2a2xx x2a2

xCCx2ax2a2x2x2a2lnx Cx2a2x2x2a250．

x2(x2a2

dx＝ x

lnx Cx2ax2a252．

dx ＝1arccosaCx x2x x2a2xx2 x2a2x2x2 x2a2x2a253．

a2x x2a2＝x2a2x x2a2＝x2x2a2

lnx Cx2a2x2a2

(x2

a2)3dx

x(2x28

5a2)

a4lnx Cx2ax2a2313(x2a2)355．x x2a13(x2a2)356．x2

x2a2dx

x(2x2

a2)

a4x2a2x2a2lnx2a2x2a257．

x2a2x

x2x2a2

8 8x2x2a2xaarccos Cx58．

x2a2dxx2

＝ x lnx Cx2x2a2a2x2

(a0)的積分a2x259．a(chǎn)2x2

＝arcsin

xaC60．61．

dx ＝ x C(a2x2)3a2 a2(a2x2)3a2 a2x2a2x2a2x262．a(chǎn)2a2x2

x(a2x2)3x2

dx＝ 1a2a2x2x a2x2

Ca2 x

arcsin C2 2 a64．

x2(a2x2)3

dx＝ x

xarcsinaCa2xa2x266．

dx ＝1x a2x a2x2x2 a2xx2 a2x2

Ca a2x2xa2a a2x2xx a2x267．a(chǎn)2x a2x22

a2 2

xaCa2x268．(a2x2)3dx＝x(5a22a2x2813(a2x2)369．x a2x2d13(a2x2)3

3 xa4arcsin C8 aa2x270．x2 a2x2dx＝x(2x2a2x28

a4 xarcsin Ca2a2x2a a2x2x71．

a2x2xx x

aln Ca2x272．a(chǎn)2x2da2x2x2 x

xarcsinaCaxax2bxcax2bxax2bxcaxax2bxcdx xaax2bxa

＝2ax2ax4aax bxc21aax21aax2bxc

(a0)的積分aa ax2bxcln2axaa ax2bxccbxcbxax277

＝

arcsin

2axb Cb2b24accbxax22ax4a2ax4acbxax28 a3

arcsin

2axb Cb24ac78 ． b24accbxax21acbxax22 a3b1acbxax22 a3b24acxxbxa(xa)(bx)79．

xadx＝(xb) (ba)ln(xaxxaxb

xb)Cxa80．xa

xabxxabxxadx＝(xb) (xabxxabx81．82

＝2arcsindxdx(xa)(bx)

C (ab)xxabx(xa)(bx)dx2xab (x2xab (xa)(bx)4xabx4（十一）含有三角函數(shù)的積分sinxdx＝cosxCcosxdx＝sinxCtanxdx＝lncosxCcotxdx＝lnsinxCsecxdx＝lntan(4

x)C＝lnsecxtanxC2x2cscxdx＝lntan C＝lncscxcotxCx2sec2＝tanxCcsc2＝cotxCsecxtanxdx＝secxCcscxcotxdx＝cscxCsin2xx＝x1sin2xC2 4cos2＝x1sin2xCnnsinnx＝1sinn1xcosx nn

xdxn nncosn＝1cosn1xsinx cosnn

xdxn n97．dx ＝

cosx

n2 dxsinn

x n1sinn1x n1 sinn2x98．

dx ＝1 sinx n2 dxcosnx n1cosn1x n

cosn2xcosmxsinnxdx

1 cosm1

xsinn1x

m1cosm2

xsinnxdxmn mn＝1 n1mncosm1xsinn1xmncosmxsinn2xdx100 ． sinaxcosbxdx ＝1 cos(ab)x 1 cos(ab)xC2(ab) 2(ab)sinaxsinbxdx＝

1 sin(ab)x 1 sin(ab)xC2(ab) 2(ab)cosaxcosbxdx＝

1 sin(ab)x 1 sin(ab)xC2(ab) 2(ab)103．104．

dx ＝ 2a2b2aa2b2atanxatanxbatanxb2b2a2b2a22

arctan

atanxba2b2a2b2

(a2b2)b2a2ab2a2

ln

(a2b2)

dx ＝abcosx

x2 aba2 abab ababab

(a2b2)106．

dx ＝ ln1 aba1 abab ba

C (a2b2)tan xtan 2x2atan xtan 2x2abaaba

b＝arctan( tanx)＝btanxabtanxbtanxabtanxa108．

dxa2cos2xb2sin2

＝12ab

ln Cxsinaxdx

1sinaxa2

1xcosaxax2sinaxdx＝1x2cosax2xsinax2cosaxCa a2 a3xcosaxdx＝1a2

cosaxa

xsinaxCx2cosaxdx＝1x2sinax2xcosax2sinaxCa a2 a3（十二）含有反三角函數(shù)的積分（其中a0）

arcsin dxxax

xarcsinx Ca2a2x2114．

xx arcsinx

x2 a2 xx a2x2＝( )arcsin x a2x2＝2 4 a 4x2

x x3 xa2x21＝arcsin dx arcsin (x22a2) a2x21＝a 3 a 9a2xa2x2arccos a

xarccos Ca117．

xx arccosx

x2 a2 xx a2x2＝( )arccos x a2x2＝2 4 a 4x2

x x3 xa2x21arccos dx＝arccos (x22a2) a2x21a 3 a 9

arctan dxxax

xarctanxxax

aln(a22

x2

)C120．x

x ax＝1(a2x2)arctan xCarctanad 2 a 2121．

x ＝x3 x a a3x2arctan a

arctan x2 ln(a2x2)C3 a 6 6（十三）含有指數(shù)函數(shù)的積分axdx

1 axClnaeaxdx＝1eaxCaxeaxdx

1(ax1)eaxa2nxneaxdx＝1xneax xn1eaxdxna axaxdx＝x ax

1 axClna (lna)2xnaxdx＝1

xnax

xn1axdxnlna lnaneaxsinbxdx＝eaxcosbxdx

1a2b21a2

eax(asinbxbcosbx)Ceax(bsinbxacosbx)eaxsi