elliptic integral SQRT(1 + (asinx)^2)dx

[dts5044]

the problem is to express the [integral from 0 to pi/2] SQRT(1 + (asinx)^2)dx where a > 0 (constant) in terms of the elliptic integral E. My first instinct was to substitute asinx = tant and work from there, but I wasn't able to nail it down. Could someone tell me if this is the right substitution to start with? And if not, what is?

 

[galactus]

\(\displaystyle \int_{0}^{\frac{\pi}{2}}\sqrt{1+a^{2}sin^{2}(x)}dx\)

This is an Elliptic integral of the 2nd kind.

\(\displaystyle E(-a,\frac{\pi}{2})\)

Is this what you mean?.

 

[Deleted member 4993]

the problem is to express the [integral from 0 to pi/2] SQRT(1 + (asinx)^2)dx where a > 0 (constant) in terms of the elliptic integral E. My first instinct was to substitute asinx = tant and work from there, but I wasn't able to nail it down. Could someone tell me if this is the right substitution to start with? And if not, what is?

Elliptic integrals cannot be solved in closed form. There are tabular solutions of this integral.

 

[dts5044]

no, I'm sorry. I need to rewrite this integral in the form SQRT(1 - (ksinx)^2)dx (which is the elliptic integral E). I don't need to solve it I just need to rewrite it in this form. And the stipulations for the elliptic integral E(k,phi) are (at least as far as I know) 0 < k < 1 and 0 <= phi <= pi/2. So Galactus' solution is only valid for -1 < a < 1. I need to rewrite it so it is valid for all cases. Is that a better explanation? Sorry for the confusion!