Integral, trig substitution

[jacks]

$\displaystyle (3)\;\; \int \frac{1}{\sqrt{1-x^4}}dx$

 

[Sudharaka]

$\displaystyle (3)\;\; \int \frac{1}{\sqrt{1-x^4}}dx$

Hi jacks,

This integral cannot be expressed in terms of elementary functions. See this.

Kind Regards,
Sudharaka.

 

[CaptainBlack]

Hi jacks,

This integral cannot be expressed in terms of elementary functions. See this.

Kind Regards,
Sudharaka.

That is strong evidence, but is not water-tight as IIRC neither Mathematica nor Alpha implements the full Risch algorithm and will occasionally drop through to special functions where an elementary integral does exist.

CB

 

[QuestForInsight]

Chebyshev's theorem: If $a, b \in\mathbb{R}$ and $m,p,n \in\mathbb{Q}$ then the (indefinite) integral of $ x^m\left(a+bx^n\right)^p$ can be written in terms of elementary functions if and only if one of $p,~ (m+1)/n, ~p+ (m+1)/n ~\in\mathbb{Z}$. In our case we have $m=0, ~ p = -\frac{1}{2}, ~ a = 1, ~ b = -1$ and $n = 4$. Clearly $p = -\frac{1}{2} \not\in\mathbb{Z}, ~ (m+1)/n = \frac{1}{4} \not\in\mathbb{Z}$ and $p+(m+1)/n = -\frac{1}{4} \not\in\mathbb{Z}.$ Thus $\int \frac{1}{\sqrt{1-x^4}}\;{dx}$ cannot be written in terms of elementary functions.