axdx a

xlnxln(lnx3.

1x1x1x1xx x 1x1212x

(x2(x2

2a x dxx(1xex

lntan cosxsin1x2 axdxaarcsinxx2aa ax

a a a

a2a2xa2x

dx

a2x2 a2x2xlnxln(ln

lnlnx211

111

d1x211 1 1 11 1111111ln13 exe ex e2x

dexe2x

arctanex411x2 1[arcsin11x2

]xxsint,t

2 cos 1x sint1 sintcostsintcost2

sintcos sintcost 21lnsintcost21arcsinxlnx2

115(x2(x21txxtant,t, 2 (x2 (x2

3dtcossintC

tan2t1162 212xu2x

,x

2u,dxu2 du 1

12 112

1uuln1u2x2x )2x2x7

dx3a2arcsinxa3a2a

2axx2 dx

2a

2axx

dx

a2(xa) a2(1sin

acos

acosudu

12sinu xxaasinu,u,,xa1sinu,dxacos22 x2asin28

12sinu

1cos2

xa1sin 32sinucos2u

sinux a23u2cosu1sin2u

uarcsinxa aua23u2cosu1sinucosuau2222

2axxa2ax

x2axa22ax 1x

2ax 2ax 9x

exxx(1xex)dxxex(1xex) d xex(1

xe

1xe

dxex ln(xex)ln(1xex)lntan dxlntan dxcosxsin

cos2xtan lntanxdtanxtanlntanxdlntanx1(lntanx)22 一 推設(shè)函數(shù)uux)和vvx)具有連續(xù)導(dǎo)數(shù)uvuvuvuvuvuvuvdxuvuvdxudvuv

xdexxexexdx(x1)exxlnxxdlnxxlnxdxx(lnx1) xarcsinxxdarcsin1xarcsinx 11xarcsinx1要點(diǎn)：適當(dāng)選擇

Ixcosucos xdx

dx2

dv Ix2cosxx2sin ux,cosxdxdsinxdvIxdsinxxsinxsinxsinxcosxC要點(diǎn)：適當(dāng)選擇Ix3lnuln x3dx 4 x3lnxdx1x4lnx 1x4lnx1x4C Ixarctan221arctanxdx222x22

arctanx

1d(arctanx)x2

x2 2x2

arctanx x

x2 arctanx Ix2exx2ex2xex

ux2,exdxdexux,exdxdvx2ex2(xexex)CIsin(lnsin(ln xsin(lnx)sin(lnxsin(lnx) xxsin(lnx)xcos(lnx)xd[cos(lnx)]sin(lnx[sin(lnx)sin(ln sin(lnx)dxx[sin(lnx)cos(lnx)]C2Iexsinexsin exsinexsinxexd(sinexsinxexcosxdxexsinxcosxdeexsinx(excosxexdcos注意循環(huán)形exsinex注意循環(huán)形exsine

sin

e(sinxcosx)Ce21I1

Ixarctanx11 arctanx1x2d(arctan1x2 arctan1x2

dx11111111111 arctanxln(11Iex u x,xeudu22udeu2ueu2eu2ueu2euC2x1ex2已fx)的一個(gè)原函數(shù)是ex,求xf2解xfx)dxxdfx)xfxf兩邊同時(shí)x求導(dǎo),

f(x)dxex2C2f(x)2xex2xf(x)dxxf(x)f(2x2ex2ex2CIsec3secxdtanxsecxtanxtanxdsecsecxtanxsecxtan2secxtanxsec 2xsec3xdxsecxtanxsec3xdxsec1secxtanxlnsecxtanx2

,a0,n (x2

a2

(

a2

(

a2

(x2a2 (x2a2

2(n2(n

(x2a2)nx2a2(

a2

(

a2 (x2a2

2(n

2a2(nn n

(2n 2a2(n

(x2a2

n1 (15)I ex(1sin(15)I ex(1 2I2

22

2 2(e

2

extanx 2[(exd(tanx)tanxdex

d(extanx extanxC2 (16)Ix2excosee

esinxdxecosxdx2

sinxcosxsinxcosxexsinxcosxdxexsinxI

x2d22

sinxcosxexe 2

sinxcosx ??2一種建議的選擇u(x)三角函

練習(xí)1、xsinxdx 2、arcsinxdx 3計(jì)算x2lnxdx，可設(shè)u,dv 4、計(jì)算excosxdx，可設(shè)u,dv 5、計(jì)算x2arctanxdx，可設(shè)u,dv 6計(jì)算xexdx可設(shè)u ,dv 2 (ln1、

2、

dxx3、eaxcosnxdx 4、e3xdxxearctan5、cos(lnx)dx 6、 3dx(1x2)x三、已知sinxfx)的原函數(shù)，求xfx)dxx四、設(shè)fx)dxFxCfx)fx)f1(x)dxxf1(x)Ff1(x)C練習(xí)題答一、1、xcosxsinxC1x2、x1x3、lnx,x2dx5、arctanx,x2dx

C4、ex cosxdx6、xe