Elementary function for n > 0 is n=1

[hedlund]

$$\int x^n \cdot \sqrt{1-x^n} \ dx$$
It seems as the only time this is an elementary function for n > 0 is n=1 and n=2, can you prove / disprove this? n is an integer

 

[dextercioby]

$$\int x^n \cdot \sqrt{1-x^n} \ dx$$
It seems as the only time this is an elementary function for n > 0 is n=1 and n=2, can you prove / disprove this? n is an integer

$$\int x^{n}\sqrt{1-x^{n}} dx=\int (-\frac{x}{n})(-nx^{n-1})\sqrt{1-x^{n}} dx=-\frac{2}{3}\frac{x}{n}(1-x^{n})^{\frac{3}{2}} +\frac{2}{3n}\int (1-x^{n})^{\frac{3}{2}} dx$$

The last integral can be solved immediately for "n=1" and through a sin/cos substitution for "n=2".As for "n>=3" (natural) it is impossible to solve analitically and express it through "elementary functions".

Daniel.

 

[daster]

As for "n>=3" (natural) it is impossible to solve analitically and express it through "elementary functions".

Can you prove that?

 

[dextercioby]

Can you prove that?

I'm not a mathematician and I'm not claiming to be one.That assertion was purely based on my mathematical "flair" and on my past experience of solving integrals.For a proof or for a counterexample i'd advise you to consult a book which extensively covers integration in general and elliptic integrals in particular.


Daniel.