Limit of a sequence defined as a definite integral Given two sequences $$a_n=\int_{0}^{1}(1-x^2)^ndx$$ and $$b_n=\int_{0}^{1}(1-x^3)^ndx$$ where $n$ is a natural number, then find the value of $$\lim_{n \to \infty}(10\sqrt[n]{a_n}+5\sqrt[n]b_n)$$
I have no starts. Looks good though. Some hints please. Thanks.

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\begin{align}
&\mbox{Asymptotically}:
\\[3mm] &\ \braces{%
\left.\int_{0}^{1}\pars{1 - x^{\mu}}^{n}\,\dd x\,
\right\vert_{\ \mu\ >\ 0}}^{1/n} \sim {\alpha^{1/n}\pars{\mu} \over n^{1/\pars{\mu n}}}\
\stackrel{n\ \to \infty}{\longrightarrow}\
\color{#f00}{\large 1}\,,\qquad \mu = 2,3.
\\[3mm] &\ \mbox{where}\ \alpha\pars{\mu} >0\,, \forall\ \mu > 0
\\[3mm] & \mbox{}
\end{align}
In another words, $\ds{\lim_{n \to \infty}a_{n}^{1/n} = \lim_{n \to \infty}b_{n}^{1/n} = \color{#f00}{\large 1}}$.

$$
\mbox{Then,}\quad
\lim_{n \to \infty}\pars{\color{#f00}{10}\,a_{n}^{1/n} + \color{#f00}{5}\,b_{n}^{1/n}} = 10 + 5 = \color{#f00}{15}
$$
