Evaluating Riemann Summation Limit: $\lim\limits_{n\to \infty}\sum\limits_{k=1}^n\frac{6(k-1)^2}{n^3}\sqrt {1+2\frac{(k-1)^3}{n^3}}$

Question

$$\lim_{n\to \infty}\sum_{k=1}^n\frac{6(k-1)^2}{n^3}\sqrt {1+2\frac{(k-1)^3}{n^3}}$$

I have no idea how to approach this problem apart from trying to convert into a definite integral using the left Riemann sum formula, but I have failed.

Answer 1

Hint: By Riemann sum we have
$$\lim_{n\to \infty}\sum_{k=1}^n\frac{6(k-1)^2}{n^3}\sqrt {1+2\frac{(k-1)^3}{n^3}} \\=6\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n\left(1-\frac{k}{n}\right)^2\sqrt {1+2\left(1-\frac{k}{n}\right)^3}\\=6\int_0^1 (1-x)^2\sqrt{1+2(1-x)^3}dx \\=6\int_0^1x^2\sqrt{1+2x^3}dx$$