# Tagged Questions

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### How do I integrate $\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}$

How do I evaluate this indefinite integral, for $|k| < 1$: $$\int\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}\mathrm{d}x$$ I tried the change of variable $t=\sin x$, and obtained two integrals, but I ...
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### Does $\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi$ have a closed form in terms of elliptic integrals?

Consider the following integral for real $a, b$ such that the square root is real: $$I=\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi$$ For $a = 0$, the integral is easily ...
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### A tricky integral (flux of a point charge through a disk)

The integrals: $$\oint \frac{r\,dr\,d\phi}{\left(L^2+r^2+h^2+2Lr\cos\phi\right)^{3/2}}\\ \oint \frac{dx\,dy}{\left((L+x)^2+y^2+h^2\right)^{3/2}}$$ If we have a point charge at the origin and we ...
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### Looking for examples where $f(z)=\operatorname{inv} \int_{0}^{z} g(z)\, dz$ with $f(z)$ entire and $g(z)$ not meromorphic.

I'm looking for examples where $f(z)=\operatorname{inv}\int_{0}^{z} g(z) \, dz$ with $f(z)$ entire and $g(z)$ not meromorphic. For clarity, by $\operatorname{inv}$, I mean the functional inverse. ...
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### Derivative of the elliptic integral of the first kind

The complete elliptic integral of the first kind is defined as $$K(k)=\int_0^{\pi/2} \frac{dx}{\sqrt{1-k^2\sin^2{x}}}$$ and the complete elliptic integral of the second kind is defined as ...
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### Incomplete elliptic integral of the second kind and the arc length of an ellipse - does a simple relation exist?

Short introduction For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I ...
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### Calculating the integral $\int_{0}^{\pi /6}\sqrt{1-\left(\frac{R_s\sin \theta }{C_L}\right)^2} d\theta$

I want to integrate $I=\int\limits_{0}^{\pi /6}{\sqrt{1-{{\left( \frac{{{R}_{s}}\sin \theta }{{{C}_{L}}} \right)}^{2}}}d \theta}$. I get incomplete elliptic integral $E(z\mid m)$ in the calculation by ...
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### Approximations of the incomplete elliptic integral of the second kind

For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
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### Help computing rational square root integral

Those integral techniques I thought I'd never need to remember are coming back to bite me. Is there a way to compute integrals of the form: $$\int_0^\infty \frac{1}{\sqrt{1 + a x^2 + b x^4}} dx$$ I ...
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### Inversion of elliptic integral

I have an equation of the type $$p=\int_0^b\sqrt{\left(a^2-x^2\right)\left(b^2-x^2\right)}dx,$$ in which $a$ and $b$ (with $a>b>0$) are (known) functions of some parameter $H$ (such that it is ...
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### Can it possible to calculate this integral directly, probably concerning ellipti integral

A student asked me to help her calculating this problem: Assume the length $L$ of a curve is given, and the equation of the curve is \begin{gather*} y(x)=A \sin\Big(\pi x-\frac{\pi}{2}\Big), 0\leq ...
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### Evaluating the average distance from a point in the unit disk to the disk

I am interested in finding the average Euclidean distance from a point $(x,y)\in\mathbb{D}_2$, the unit disk $\{(u,v):u^2+v^2\leq 1\}\subseteq\mathbb{R}^2$, to the disk $\mathbb{D}_2$. This amounts to ...
### Can this integral $\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$ be written in the form of a elliptic integral
I am trying to find the magnetic field due to an elliptic loop of wire. How to do integrals of the type $$\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$$ Where a and b are ...