∫0+∞x/(1+x)3dx

∫₀^+∞x/(1+x)³dx
【正确答案】：∫₀^+∞[x/(1+x)³]dx=lim_b→+∞∫₀^b(1+x-1)/(1+x)³dx =lim_b→+∞[∫₀^b[1/(1+x²)]d(1+x)-∫₀^b[1/(1+x)³]d(1+x)] =lim_b→+∞[-[1/(1+x)]|₀^b+1/2•1/(1+x)²|₀^b =lim_b→+∞{1-1/(1+b)+1/2•[1/(1+b)²]-1/2}=1/2