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?

  • $\begingroup$ Cool. I didn't know this rule of thumb. Wish I did. Anyway, your question seems to be kind of weird. You're asking for a counterexample under specific restrictions right? Hence, I think your question ought to be rephrased $\endgroup$
    – BCLC
    Jan 21 '16 at 7:21
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    $\begingroup$ That's right. And I think I phrased the question adequately. $\endgroup$ Jan 21 '16 at 7:22
  • $\begingroup$ I guess if you didn't know this rule of thumb it would seem an odd question, but I think it is pretty well known and used. At any rate the fact that we can't think of a counterexample off the top of our heads seems motivation enough. $\endgroup$
    – Elliot G
    Jan 21 '16 at 8:00
  • $\begingroup$ @CarlHeckman I mean the title question $\endgroup$
    – BCLC
    Jan 21 '16 at 8:11
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    $\begingroup$ This isn't about why LIATE is wrong, but a possible problem with using it in teaching. This method doesn't encourage much thinking. Many times students will follow the rule blindly without understanding why they make the choice they do. In contrast, if you talk about how repeated differentiation brings polynomials to zero, or how trig derivatives repeat so you can relate the current integral to a later integral, if the students grasp this, they have a better understanding than if they memorize LIATE.(note, acronyms like SOHCAHTOA I think are okay since they are about definitions not methods.) $\endgroup$ Jan 21 '16 at 11:42

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

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    $\begingroup$ Once again, I'm not particularly happy with these examples, since $\sin(x^2)$ isn't "really" a trig function (the presence of composition suggests that substitution be used instead); I would consider your second example closer to what I'm looking for (even though an example where "algebraic" means "power of x" would be better). ... However, I think your first example is important enough to mention next class. $\endgroup$ Jan 22 '16 at 6:41
  • $\begingroup$ I just noticed that you need some parentheses around $1+x$. Also, the integral is easier to do if you let $u=xe^x$ and $\displaystyle v'={1\over (1+x)^2}$. $\endgroup$ Jan 24 '16 at 7:33
  • $\begingroup$ @CarlHeckman Why don't you state what you consider or don't consider to be trig/exp functions in OP? $\endgroup$
    – BCLC
    Jan 24 '16 at 8:08
  • $\begingroup$ Actually, I did do this for exponential functions, just not trig functions. $\endgroup$ Jan 25 '16 at 4:48
  • $\begingroup$ For the latter, I think it is easier to notice the inherent quotient rule (I think I've seen this in a MIT Integration Bee) $$\int \frac{xe^x}{(x+1)^2}dx = \int \frac{(x+1)e^x - e^x}{(x+1)^2}dx$$ $\endgroup$
    – John Lou
    Feb 28 '18 at 3:43

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.


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