$$\int \sqrt{\tan x} dx$$
\begin{align}
&\int \sqrt{\tan x} dx\\
& = \int \sqrt{\frac{\sin x}{\cos x}}dx\\
& = \int \sqrt{\frac{(\cos x)'}{-\cos x}}dx\\
& = \int \sqrt{-\frac{f'}{f}}dx\\
& = \int \sqrt{-(\ln(f))'}dx\\
& = \int \sqrt{(\ln(\frac{1}{f}))'}dx\\
\text{Let }
u^2 & =(\ln(\frac{1}{f}))'\\
2udu & =(\ln(\frac{1}{f}))''dx=2uu'dx\implies \frac{du}{u'}=dx\\
& \int \sqrt{(\ln(\frac{1}{f}))'}dx\\
& = \int \frac{\sqrt{u^2}}{u'} du= \int \frac{|u|}{u'} du\\
\end{align}
My suspicion is confirmed: Had $u=\tan x$ and $u'=\sec^2 x$ does not have the trigonometric identity $1+\tan^2x=\sec^2 x$, …

Quantum mechanically, those chemistry terms are linked as follows:

1. Shells = principle quantum number n, gives the energies of the orbitals (not related to its shape)
2. Subshell = azimuthal quantum number/angular momentum quantum number l, responsible for the spherical, dumbell, clover, flower like shapes of s,p,d,f orbitals. Their orientation in space is controlled by the magnetic quantum number m.
3. Orbitals = probability of finding electrons in a region. This is done by squaring the wavefunctions with different n,l,m. The boundary is defined to be where you will find electrons 90-95…