# Elementary function for n > 0 is n=1

1. Dec 20, 2004

### 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

2. Dec 21, 2004

### dextercioby

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

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.

3. Dec 21, 2004

### daster

Can you prove that?

4. Dec 21, 2004

### dextercioby

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.