Question

大学微积分问题

题目如图，求关于x趋于正无穷y的值，题目上说不用解这个微分方程就能得出答案，但是我看不出来，😭

Answer 1

I(m,n)=1/(n+1)*∫(0,1) (lnx)^m d[x^(n+1)] =1/(n+1)*(lnx)^m*x^(n+1)|(0,1)-m/(n+1)*∫(0,1) x^n*(lnx)^(m-1) dx =-1/(n+1)*lim(x->0+) (lnx)^m*x^(n+1)-m/(n+1)*I(m-1,n) =-m/(n+1)*I(m-1,n) =[-m/(n+1)]*[-(m-1)/(n+1)]*I(m-2,n) ...... =m!*[-1/(n+1)]^m*I(0,n) =m!*[-1/(n+1)]^m*∫(0,1) x^n dx =m!*[-1/(n+1)]^m*[1/(n+1)]*x^(n+1)|(0,1) =(-1)^m*m!/(n+1)^(m+1)