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juantheron
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$\displaystyle \int\sqrt{\frac{x^2+1}{x^2-1}}dx$
topsquark said:(sigh) I can't get this thing out of my mind and I keep thinking in circles. This'll be my last post on the topic. I promise!
Okay, to be specific about the beef I originally had with this problem is that Mathematica says this is an elliptic integral of the second kind. But elliptic integrals are definite integrals. Is it possible that this indefinite integral might have a closed form solution even though the definite integral does not.
-Dan
The obvious implication is that the elliptic integral must be imaginary. But by its definition I don't think that the elliptic integral can be anything but real?chisigma said:That's very interesting the results supplied by 'Monster Wolfram'...
Wolfram Mathematica Online Integrator
Apart the 'odd identities'...$\displaystyle \frac{\sqrt{x^{2}+1}}{\sqrt{x^{2}+1}}=1$
$\displaystyle \frac{\sqrt{1-x^{2}}}{\sqrt{x^{2}-1}}=i$
... the result seems to be...
$\displaystyle \int \sqrt{\frac{x^{2}+1}{x^{2}-1}}\ dx = i\ \text{E}\ (\sin^{-1} x|-1) + c $ (1)
... and also in this case the imaginary unit appears... what's Your opinion about that?... Kind regards $\chi$ $\sigma$
Notice that the denominator of the fraction is negative unless $|x|>1$. So you should only expect to get a real value for the integral if you avoid the interval [-1,1]. In particular, the substitution $x=\cos t$ implicitly assumes that $|x|\leqslant1$ and hence inevitably leads to a non-real answer.jacks said:$\displaystyle \int\sqrt{\frac{x^2+1}{x^2-1}}dx$
The integral of $\sqrt{\frac{x^2+1}{x^2-1}}$ is a mathematical expression that represents the area under the curve of the function $\sqrt{\frac{x^2+1}{x^2-1}}$ between two points on the x-axis.
The integral of $\sqrt{\frac{x^2+1}{x^2-1}}$ has many applications in various fields of science and engineering. It is used to calculate the total displacement, velocity, and acceleration of an object over a given time period. It is also used in calculating the work done by a variable force and the total energy of a system.
The steps to solve the integral of $\sqrt{\frac{x^2+1}{x^2-1}}$ are as follows:
Yes, the integral of $\sqrt{\frac{x^2+1}{x^2-1}}$ can also be solved using other methods such as integration by parts, trigonometric substitution, and partial fractions. However, the substitution method is the most commonly used method for solving this integral.
Yes, when the limits of integration are infinite, or when the limits are not defined, the integral of $\sqrt{\frac{x^2+1}{x^2-1}}$ may yield an infinite or undefined result. In such cases, the integral is said to be improper, and special techniques must be used to solve it.