高數(shù)課件利用兩個(gè)函數(shù)乘積的求導(dǎo)法則

例1求積分xcosxdx解（一）令ucos

xdx=1dx2= xxcosxdx=2cosx+2sinv解（二）令u= cosxdx=dsinx=xcosxdx=xdsinx=xsinx-sin=xsinx+cosx+C例2求積分x2exdx.解1選u=e v'=x2,則：du=e

v'dx=x2dx=1dx3原式1x3ex－1x3dex＝1x3ex－1x3exdx

u=x2 exdx=dex=x2exdx=x2ex-2xexdx=x2ex-2(xex-ex)+C

exdx=就考慮設(shè)冪函數(shù)u,使其降冪一次(假定冪指數(shù)是正整例3求積分xarctan解1令u=x,dv=arctanx 則du=dx,v=2無法進(jìn)行求解！2 xdx=dx2x

=xarctan

2arctanx-

xd(arctan222=x2arctanx-

dx

xarctanx

1(1

1+x

1+x=2arctanx-2(x-arctanx)+C例4求積分x3ln解1令ux3dvln則du=3x2,v=dv=lnxdx= 無法進(jìn)行求解解u=ln x3dx= =4 x3lnxdx=1x4lnx- =1x4lnx-1

+C慮設(shè)對(duì)數(shù)函數(shù)或反三角函數(shù)為u.使得量變繁、質(zhì)變簡！uulnx(或arccos例6exsinuexsin =sinxdex=exsinx-exd(sin=exsinx-excosxdx=exsinx-cos=exsinx-(excosx-exdcos

=ex(sinx-cosx)x

exsin

\exsinxdx= (sinx-cosx)+C2例6exsinexsinxdxsinxdexexsinxexd(sin=exsinx-excos -x=-xd(e=-excosx+excos e

sinxdxx

2(sinx-cosx)+C

cos

=(sinx+cosx)+C2sin(ln例5sin(ln解

=xsin(lnx)-xd[sin(ln=xsin(lnx)-

xcos(lnx) 1x=xsin(lnx)-xcos(lnx)+xd[cos(ln\sin(lnx)dx=x[sin(lnx)-cos(lnx)]+C2.

例7

ln(x-b)x-解：原式＝ln(x-b)d- ＝ln(x-b)-x- x-

+1 dln(x-b)x-a＝ln(x-b)-

+ 1 dxx- x- x-＝ln(x-b)- +

- 1 dxx-

x- x

ba-x-x-＝ln(x- - +x-x-x- a-解解例11求積分e3 x解：令t＝3x，則：xt3dx3t原式＝et3t2dt＝3t2det3t2etet6t3x=3t2et-6tdet=3t2et-t-etdt)=3t2et3x

-6tet+et+

2e3x-63xe3x+e3x+

1+x2(1+x2 (1+x2)dt=t(1+x2uv'dx=udv=dv+c=v+-v+例13求積 解1：選uln(1exdu

e

dv=e-xdx=-de-1+e e

e1+

xdx=解2：選uln(1

x),du

1+ex

dv=e-

=d-e-

+選c=-1,則：dv=d-e-=d 1+e 1+e e

1+e

xdx= ln(1+ex)+x+e e

1+ e例13

ex(tgx+1sec2x 解：原式＝tgxde

+e 1 xdx＝tgxdex+e dtg ex1sinx1+cos

＝d(tg ex)＝tg ex+ 'Q sin '

cos1+cosx-sinx(-cos

sin = = 1+cosx

1+cos

1+cos

1+cosx

1+cos\原式＝sin dex＋exdsinx 1+cos 1+cos

＝sin ex+c1+cos1+x21+x21+￠解 (1+x1+￠

xarctan

=

1+x21+x21+x21+x21+x2 arctanx1+x2

1+x21+1+x211+x21

xtan1+

dx=

sec2tdt=sec1+x2=ln(sect+tant)+C=ln(1+x2

)+1+x21+x2\1+x21+x2

11+x2

)+C28fx)的一個(gè)原函數(shù)是e-2

,求xfx)dxxfx)dxxdfx)xfxf=f(x)dx=

f(

\f(x)dx=e- +C

f(x)=-2xe-x\xf(x)dx=xf(x)-f( =-2x2e-x-e-x+Cu,vuvdxuvdx excosu1cosexcosxdx=excosx+exsinu2sin

1、xsinxdx 2、arcsinxdx 3計(jì)算x2lnxdx，可設(shè)u=,dv=；4、計(jì)算e-xcosxdx，可設(shè)u=,dv=；5、計(jì)算x2arctanxdx，可設(shè)u=,dv=；6計(jì)算xe-xdx可設(shè)u= ,dv=. (lnx 1、x2cos2dx

3、eaxcosnxdx 4、e3xdxxearctan5、cos(lnx)dx； 6、 3dx.(1+x2)2三、已知sinxfx)的原函數(shù)，求xfx)dxx四、設(shè)fx)dxFxCfx)fx)的反f-1x)存在，則f-1(x)dx=xf-1(x)-Ff-1(+C一、1、xcosxsinxC2、xarcsinx3、lnx,x2dx 4、e-x cosxdx5、arctanx,x2dx 6、x,e-x