$$ \begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\ &=33 \cdot \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right] \end{aligned}$$, then the integral value of $$\mathrm{k}$$ is equal to _____________

Let $$f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx} $$. If $$f(1) + f'(1) = \alpha e - {1 \over 6}$$, then the value of 150$$\alpha$$ is equal to ___________.

The integral $${{24} \over \pi }\int_0^{\sqrt 2 } {{{(2 - {x^2})dx} \over {(2 + {x^2})\sqrt {4 + {x^4}} }}} $$ is equal to ____________.

Let f(x) = max {|x + 1|, |x + 2|, ....., |x + 5|}. Then $$\int\limits_{ - 6}^0 {f(x)dx} $$ is equal to __________.