Calculate an integral depending on n Is there a way (simple or not) to calculate the following integral?
$$\int_{-1}^{1} \sqrt[n]{1-x^n} dx$$
Thanks
 A: *

*For even values of n, the simple substitution $t=x^n$ will automatically yield the beta function, which is basically a generalization of binomial coefficients.

*For odd values of n, the same basic substitution will yield a sum of complete and incomplete beta functions. The latter can also be expressed in terms of hypergeometric functions, by expanding the integrand into its binomial series, and then switching the order of summation and integration. This last trick might also cover the “numerical methods” that you seem to mention in your post.
A: $2 \text{B}(1/n, 1/n+1)/n$ where $\text{B}$ is the Beta function, if $n$ is even.
A: As already said by uranix, the result involves the hypergeometric function. Without any restriction about $n$, the formula is given by $$\int_{-1}^{1} \sqrt[n]{1-x^n} \,dx=\, _2F_1\left(-\frac{1}{n},\frac{1}{n};1+\frac{1}{n};(-1)^n\right)+\sqrt{\pi }\frac{
  \Gamma \left(1+\frac{1}{n}\right)}{\Gamma
   \left(\frac{1}{2}+\frac{1}{n}\right)} 4^{-1/n}$$ which is not the nicest form we could dream about.
For the case where $n$ is even, Robert Israel already gave the answer $$\int_{-1}^{1} \sqrt[n]{1-x^n} \,dx=\frac{2}{n}B\left(\frac{1}{n},1+\frac{1}{n}\right)$$
