Problem - 4571
Without explicitly evaluating the integral, show that
$$\lim_{n\to\infty}\int_1^2\ln^n{x}dx =0\quad\text{and}\quad\lim_{n\to\infty}\int_2^3\ln^n{x}dx = \infty$$
1) $$0 \le \lim_{n\to\infty}\int_1^2 \ln^n x dx \le \lim_{n\to\infty}\int_1^2 \ln^n 2 dx =0 $$
2) $$\lim_{n\to\infty}\int_2^3\ln^n{x}dx > \lim_{n\to\infty}\int_{1.00001e}^3\ln^n{x}dx > \lim_{n\to\infty}\int_{1.00001e}^3\ln^n{(1.00001e})dx = \lim_{n\to\infty}\int_{1.00001e}^31.00001^ndx $$