Integral, trig substitution

jacks

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

Sudharaka

Well-known member
MHB Math Helper
$\displaystyle (3)\;\; \int \frac{1}{\sqrt{1-x^4}}dx$
Hi jacks,

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

Kind Regards,
Sudharaka.

CaptainBlack

Well-known member
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

Member
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.