A high number problem of limit I = ∫ x ^ n / 1 + X DX the limit of I The upper and lower limits of the integral are 0 and 1

A high number problem of limit I = ∫ x ^ n / 1 + X DX the limit of I The upper and lower limits of the integral are 0 and 1

x^n/(x+1)=x^(n-1)-x^(n-2)+---+(-1)^(n-1)+(-1)^n/(x+1)
∴I=∫x^n/(x+1) dx
=[x ^ n / N-X ^ (n-1) / (n-1) + --- + (- 1) ^ (n-1) x + (- 1) ^ Nin | x + 1 |] (0,1) (upper and lower limits)
=1/n-1/(n-1)+---+(-1)^(n-1)+(-1)^nIn2
=(-1)^(n-1)[1-1/2+1/3+---+(-1)^(n-1)/n]+(-1)^nIn2
In (x + 1) = x-x ^ 2 / 2 + x ^ 3 / 3 + --- + (- 1) ^ (n-1) x ^ n / n+---
Let x = 1, then in2 = 1-1 / 2 + 1 / 3 + --- + (- 1) ^ (n-1) / n+---
∴limI=-(-1)^nIn2+(-1)^nIn2=0