When is LIATE simply wrong? 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?
 A: A couple of them I found in a blog post, summarizing here: $$\int{x^3\sin{x^2}dx}$$
Here $u=x^3$ which you could choose based on LIATE does not work since it is hard (if not impossible) to calculate the antiderivative of $\sin{x^2}$. The 'correct' choice would be $u=x^2$ so that $dv=x\sin{x^2}$, which does work.
Or $$\int\frac{xe^x}{(1+x)^2}dx$$
With the LIATE rule you would try something like $u=\frac{x}{(1+x)^2}$ with $dv=e^xdx$ which would require you to calculate $\int e^x\frac{(1+x^2)-2(1+x)x}{(1+x)^4}dx$. The 'correct' choice here would be $u=e^x$ and $dv=\frac{x}{(1+x)^2}dx$, and with $w=1+x$:
$$v=\int\frac{x}{(1+x)^2}dx=\int\frac{w-1}{w^2}dw=\log(1+x)+\frac{1}{1+x}\\
\int\frac{xe^x}{1+x^2}dx=e^x(\log(1+x)+\frac{1}{1+x})-\int{(\log(1+x)+\frac{1}{1+x})e^xdx}\\
=e^x(\log(1+x)+\frac{1}{1+x})-\log(1+x)e^x+C\\
=\frac{e^x}{1+x}+C
$$
Source: https://mathnow.wordpress.com/2009/10/14/liate-ilate-and-detail
A: 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.
