TITLE: When is LIATE simply wrong?
QUESTION [18 upvotes]: I'm currently teaching Calculus II, and yesterday I covered integration by parts and mentioned the LIATE rule. I also gave the usual "it works 99% of the time", but started wondering whether there are any cases where LIATE simply gets the choice of $u$ and $v'$ wrong.
(For those of you who don't know what LIATE is, check out https://en.wikipedia.org/wiki/Integration_by_parts#LIATE_rule )
I don't consider the example listed at the link above to be what I'm looking for, because I don't consider $e^{x^2}$ to be an exponential function here (only $a^{bx}$). (I don't consider $\tan x$ to be a "trig" function, either, in this context.)
Does anyone have a "pet" example that they show?

REPLY [4 votes]: My favorite example of LIATE failure is:
$$\int x^{13}e^{\left(x^7\right)}\;dx$$
What works is:
$$f = x^7$$
$$dg = x^6e^{\left(x^7\right)}\;dx$$
$$df = \frac{1}{7}x^6$$
$$g = \frac{1}{7}e^{\left(x^7\right)}$$
I ran this through https://www.integral-calculator.com/, and it just substituted $u=x^7$ right at the start, resulting in an easier problem, in which LIATE works.