微积分自学考试试题及答案

1、(2x-1)/(x^2+3x+2)=(2x-1)/[(x+1)(x+2)]=-3/(x+1)+5/(x+2)∴∫(2x-1)/(x^2+3x+2)=∫[-3/(x+1)+5/(x+2)]dx=∫[-3/(x+1)]d(x+1)+∫[5/(x+2)]d(x+2)=-3ln|x+1|+5ln|x+2|+C 2、利用变限积分求导公式：d[∫f(t)dt]/dx=f[φ(x)]*φ'(x)d[∫<0,x^2>sintdt]/dx=sin(x^2)*2x=2xsin(x^2)∴lim[(∫<0,x^2>sintdt)/x] x->0=lim{d[∫<0,x^2>sintdt]/dx}/(dx/dx) 洛必达法则=lim[2xsin(x^2)/1]=0 3、∫<1,+∞>1/x^4*dx=∫<1,+∞>x^(-4)dx=<1,+∞>[-1/3*x^(-3)]=-1/3-0=-1/3 4、5两题用二元函数极限的洛必达法则lim[f(x,y)/g(x,y)]=lim{[f'x(x,y)dx+f'y(x,y)dy]/[g'x(x,y)dx+g'y(x,y)dy]}其中，极限点为 x->x0, y->y0，且dx=x-x0, dy=y-y0第4题：lim{[√(xy+4)-2]/(xy)} x->1, y->0f(x,y)=√(xy+4)-2, g(x,y)=xy; x0=1, y0=0; dx=x-1, dy=yf'x(x,y)=1/2*y/√(xy+4), f'y(x,y)=1/2*x/√(xy+4)g'x(x,y)=y, g'y(x,y)=x∴lim{[√(xy+4)-2]/(xy)} x->1, y->0=lim{[1/2*y/√(xy+4)*dx+1/2*x/√(xy+4)*dy]/[y*dx+x*dy]}=lim[1/2*1/√(xy+4)]*{[y(x-1)+xy]/[y(x-1)+xy]}=1/4*lim{[2xy-y]/[2xy-y]}=1/4第5题：lim[sin(xy)/y] x->1, y->0f(x,y)=sin(xy), g(x,y)=y; x0=1, y0=0; dx=x-1, dy=yf'x(x,y)=ycos(xy), f'y(x,y)=xcos(xy)g'x(x,y)=0, g'y(x,y)=1∴lim[sin(xy)/y] x->1, y->0=lim{[ycos(xy)*dx+xcos(xy)*dy]/[0*dx+1*dy]}=limcos(xy)*{[y(x-1)+xy]/[0+y]}=lim1*{[2xy-y]/y}=lim(2x-1)*lim(y/y)=(2-1)*1=1

吃饭上课。晚上做。