Answer: $B$
$$\underset{n \to \infty}{\lim} \int^1_0x^n\ln(1+x)dx$$
can be simplified to:
$$\ln(2)-\underset{n \to \infty}{\lim}\int^1_0\frac{x^{n+1}}{1 + x}dx$$
$$\because \bigg |\int^1_0\frac{x^{n+1}}{1+x}dx\bigg | \le \int^1_0 \big |x^{n+1} \big|dx$$
Now, here the bound $1+x$ is always greater than equal to $1$
$\Rightarrow \frac{1}{1+x} \le1$
$$\therefore \underset{n \to \infty}{\lim}\int^1_0 x^n \ln(1+x)dx = \ln(2)$$
Thus, $B$ be is the correct option.