I was trying to evaluate the integral $$\int_0^1\sqrt[4]{1-x^4}\,\mathrm dx$$ but failed to do so in terms of elementary functions. I wondered what happens more generally; that is, for some integer $m>1,$ is the quantity $$\int_0^1 \sqrt[{2m}]{1-x^{2m}} \, \mathrm dx$$ expressible in terms of elementary functions alone?
I was moved to consider such integrals because of their relationship to the case $$\int_0^1\sqrt{1-x^2}\,\mathrm dx=\fracπ4,$$ which is related to the unit circle in the euclidean plane $x^2+y^2=1.$ I then wondered whether the theory of circular functions was sufficient to take care of the related closed curves $$x^{2m}+y^{2m}=1,$$ with $m$ as above. In particular I wondered whether their areas could be expressed in terms of the areas of the unit circle, hence the above integrals, and question.
Thanks.
Let $ m\in\mathbb{N}^{*} \cdot $
Well, using the substitution $ \left\lbrace\begin{aligned}t&=x^{2m}\\ \mathrm{d}x&=\frac{1}{2m}t^{\frac{1}{2m}-1}\,\mathrm{d}t\end{aligned}\right. $, we get : \begin{aligned} \int_{0}^{1}{\sqrt[2m]{1-x^{2m}}\,\mathrm{d}x}=\frac{1}{2m}\int_{0}^{1}{t^{\frac{1}{2m}-1}\left(1-t\right)^{\frac{1}{2m}}\,\mathrm{d}t}=B\left(\frac{1}{2m},1+\frac{1}{2m}\right)&=\frac{\Gamma\left(\frac{1}{2m}\right)\Gamma\left(1+\frac{1}{2m}\right)}{2m\Gamma\left(1+\frac{1}{m}\right)}\\&=\frac{\Gamma^{2}\left(\frac{1}{2m}\right)}{4m\Gamma\left(\frac{1}{m}\right)} \end{aligned}