I would like to ask the following two:

- For the integral: \begin{equation} \int_{0}^{t}\left( 1-s^p \right)^{\frac{1-p}{p}}ds \end{equation} I know that it is reduced to the following product involving the hypergeometric function: \begin{equation} \int_{0}^{t}\left( 1-s^p \right)^{\frac{1-p}{p}}ds={}_{2}F_{1}\left(1-\frac{1}{p}, \frac{1}{p}; 1+\frac{1}{p};t^p \right)t \end{equation} I am not yet able to prove that, but my thoughts are to expand $\left( 1-s^p \right)^{\frac{1-p}{p}}$ using the binomial theorem, then use that to approximate the series for the hypergeometric function mentioned.
- Also, for the following: \begin{equation} x={}_{2}F_{1}\left(1-\frac{1}{p}, \frac{1}{p}; 1+\frac{1}{p};t^p \right)t \end{equation} I would like to ask whether it is possible to solve explicitly for $t$. ${}_{2}F_{1}(a,b;c;z)$ is the Gauss ordinary hypergeometric function and $p>1$. Since I am not familiar with this kind of manipulations, I would like to ask if there is any transformation for this kind of functions, which would help in this situation. Thanks!