calculating the limit $\lim_{n\to ∞} (n+1)\int_{0}^1 x^nln(1+x)dx$

Question

$$\lim_{n\to ∞} (n+1)\int_{0}^1 x^nln(1+x)dx$$

there are four options (A) 0 (B) ln 2 (C) ln 3 (D) ∞.

my answer is '0' . is that correct ?

Answer 1

Hints:

• With a basic integration by parts, you get $(n+1)\int_0^1 x^n \ln(1+x)dx = \ln 2 - \int_0^1 \frac{x^{n+1}}{1+x}dx$.
• for $0<x<1$, $0 < \frac{x^{n+1}}{1+x} < x^{n+1}$

What do you think now?