I've just started studying improper integrals and I've come across this one:
$$\int_{0}^{\infty}{\sqrt{\tan{x}}dx}$$
The main issue I have with this is that the root is imaginary when tanx is negative, so I don't know how to deal with this.
Presumably to deal with the infinity in the interval, we take some limit like this:
$$\lim_{t\rightarrow\infty}{\int_{0}^{t}{\sqrt{\tan{x}}}dx}$$
But even this feels illogical until I find a way of quantifying the integral of the complex part of this graph. I suspect there is a way using substitutions and whatnot to find a solution to the general integral of $\sqrt{\tan{x}}$, but presumably this will contain $\tan{x}$ in the solution, and (to my knowledge) there is no way to evaluate anything containing a $\tan{\infty}$. I have worked with $\sin{\infty}$ and $\cos{\infty}$ before in the sense that they are strictly between $-1$ and $1$, and so in certain situations an expression containing them can be evaluated, but in my (admittedly limited) experience $\tan{\infty}$ cannot be dealt with in the same way.
I have also considered making use of polar coordinates and taking a double integral, in the standard way that the Gaussian integral is computed, but this was even more of a stretch as I do not think the tan function would interact well with itself in the way the Gaussian function does.
Any help is welcome, and if someone could also explain what it actually means to take an integral of a function that has no real value in the interval that would also be very helpful.
