∫0～∞x/(1+x)∧3dx=
问题描述：
∫0～∞x/(1+x)∧3dx=
答
∫xdx/(1+x)³=lim(a->∞)∫xdx/(1+x)³ (应用广义积分定义)
=lim(a->∞)∫[(1+x)-1]dx/(1+x)³
=lim(a->∞)∫[1/(1+x)²-1/(1+x)³]dx
=lim(a->∞)[(1/2)/(1+x)²-1/(1+x)]│
=lim(a->∞)[(1/2)/(1+a)²-1/(1+a)-1/2+1]
=lim(a->∞)[(1/2)(a/(1+a))²]
=lim(a->∞)[(1/2)(1/(1/a+1))²]
=1/2.