Evaluate the limit: $\mathop {\lim}\limits_{n \to \infty } \frac{\pi }{2n} \left( 1 + \cos \frac{\pi }{2n} + \cos \frac{2\pi }{2n} + \dots + \cos \frac{(n - 1)\pi }{2n} \right)$

Let $[ \cdot ]$ denote the greatest integer function and $f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum_{j=1}^{\infty} f(j)$ is equal to ........... .

The value of $\lim_{n \to \infty} \left[ \frac{1}{n}\sin \left( \frac{1}{n} \right)\left( \cos \left( \frac{1}{n} \right) \right)^2 + \frac{1}{n}\sin \left( \frac{2}{n} \right)\left( \cos \left( \frac{2}{n} \right) \right)^2 + \dots + \frac{1}{n}(\sin 1)(\cos 1)^2 \right]$ is

$\operatorname{Lim}_{n \rightarrow \infty} \frac{\pi}{2 n}\left[\sin \frac{\pi}{2 n}+\sin \frac{2 \pi}{2 n}+\ldots+\sin \frac{\pi}{2}\right]=$

Evaluate $\int_2^3 x^2 dx$ as the limit of a sum.

Evaluate the following definite integral as the limit of a sum:
$\int_{0}^{4} (x + e^{2x}) \, dx$