积分题,用变换1/x=t或1/(x+1)=t,

∫dx/[x(x+1)^3]
=∫[(x+1)-x]dx/[x(x+1)^3]
=∫dx/x(x+1)^2-∫dx/(x+1)^3
=∫dx/[x(x+1)]-dx/(x+1)^2+(1/4)(1+x)^(-4)
=∫dx/x-∫dx/(x+1)+(1/3)(x+1)^(-3)+(1/4)(1+x)^(-4)
=lnx-ln(x+1)+1/[3(1+x)^3]+1/[4(1+x)^4]
∫[0,+∞] dx/[x(x+1)^3]=1/3+1/4=7/12