LIATE : How does it work?

When doing Integration By Parts, I know that using LIATE can be a useful guide most of the time.

For those not familiar, LIATE is a guide to help you decide which term to differentiate and which term to integrate. L = Log, I = Inverse Trig, A = Algebraic, T = Trigonometric, E = Exponential.

The term closer to E is the term usually integrated and the term closer to L is the term that is usually differentiated.

Example, if I were to compute $\int x \ln x dx$ then I would be differentiating the $\ln x$ and integrating the $x$.

My question here is: Why does the 'guide' work so well? Is there a Mathematical reason behind all this?

Logs and inverse trig functions, when part of a more complicated integral, tend to cause problems. By differentiating them, you get 'simpler' more algebraic expressions, because $$\frac{d}{dx}\ln x=\frac{1}{x}$$ and $$\frac{d}{dx}\sin^{-1}x=\frac{1}{\sqrt{1-x^2}}$$ (the other inverse trig functions have similar derivatives with square roots in them). If you try to integrate them you will just create a giant mess that's harder to solve than what you started with.

On the other end, regular trig functions and exponential functions basically don't change when you integrate (or differentiate) them, so usually it is helpful to integrate them if differentiating simplifies something else. For example, integrating $e^x$ just returns $e^x$ over and over again, and integrating $\sin x$ returns the repeating sequence $-\cos x,-\sin x,\cos x,\sin x,...$

By "algebraic" I'm assuming the middle term if referring to roughly "polynomials". Polynomials are easy to differentiate or integrate, but they do change. In something like $$\int{x^3\cos x\;dx}$$ the cosine term, as we said above, can be integrated. So we might as well differentiate away the $x^3$ term repeatedly until it becomes a constant and we are left with an easy trig integral.

But if we have $$\int{x^3\ln x\;dx}$$ the logarithm is not easy to integrate. We can differentiate it to get something algebraic - so in this case integrating the algebraic part will work fine, because we will end up with an integral that's easy to solve.

Please note that although I have seen the term LIATE before, I have never used it in my life and I wouldn't have been able to tell you what it stood for. Once you get more comfortable with integration by parts you should see how to make choices in terms of what to differentiate and what to integrate, and then the rule will seem rather obvious. However, I wouldn't be overly reliant on "tricks" like this, because tricks always have exceptions. Instead I would just try to look at what "makes sense" for any given integral - and it will usually be what LIATE would tell you to do.