Calculate these integrals I want to know how to calculate any of these integrals, which arise from computing the perimeter of the unit ball in the $p$-norm.


*

*$$\int_0^1(1+p^pt^{p(p-1)})^{1/p} dt;$$

*$$\int_0^{\pi/2}(\cos(t)^{2-p}\sin(t)^2+\sin(t)^{2-p}\cos(t)^2)^{1/p} dt.$$
In the first one, I tried to expand $(1+p^pt^{p(p-1)})^{1/p}$ as
$$\sum_{k =0}^\infty \binom{1/p}{k}(pt^{p(p-1)})^k,$$
where $\binom{1/p}{k} = \frac{(1/p)\cdots((1/p)-k+1)}{k!}$.
Since
$$\int_0^1 t^{p(p-1)k} dt = \frac{t^{1+p(p-1)k}}{1+p(p-1)k} = \frac{1}{1+p(p-1)k},$$
it follows that
$$\int_0^1(1+p^pt^{p(p-1)})^{1/p} dt = \sum_{0}^\infty \binom{1/p}{k}\frac{p^{pk}}{1+p(p-1)k}.$$
I'm not sure if this converges and how to calculated it. I don't know how to proceed in the second one.
 A: Concerning the first question (excluding the trivial cases $p=0$ and $p=1$) 
$$I_p=\int\left(1+p^p t^{(p-1) p}\right)^{\frac{1}{p}}\,dt=t\,\, _2F_1\left(-\frac{1}{p},\frac{1}{(p-1) p};1+\frac{1}{(p-1) p};-p^p t^{(p-1) p}\right)$$ where appears the Gaussian or ordinary hypergeometric function (have a look here). So,
$$J_p=\int_0^1\left(1+p^p t^{(p-1) p}\right)^{\frac{1}{p}}\,dt=\, _2F_1\left(-\frac{1}{p},\frac{1}{(p-1) p};\frac{p^2-p+1}{(p-1) p};-p^p\right)$$ 
What is interesting is that $J_p$ goes through a minimum value as shown in the table below
$$\left(
\begin{array}{cc}
 1 & 2.00000 \\
 2 & 1.47894 \\
 3 & 1.45510 \\
 4 & 1.49983 \\
 5 & 1.54817 \\
 6 & 1.58972 \\
 7 & 1.62429 \\
 8 & 1.65314 \\
 9 & 1.67752 \\
 10 & 1.69837
\end{array}
\right)$$
Remarkable is $J_2=\frac{10+\sqrt{5} \sinh ^{-1}(2)}{4 \sqrt{5}}$.
For the second integral
$$K_p=\int_0^{\pi/2}\left(\cos(t)^{2-p}\sin(t)^2+\sin(t)^{2-p}\cos(t)^2\right)^{\frac 1p}\, dt$$ I have not been able to get anything analytical beside $K_1=\frac 23$ and $K_2=\frac \pi 2$. Numerical integration shows that $K_p \sim p-\frac 12$
