Question about Integration by Parts (LIATE)

Question

I know that when you do Integration by Parts, you usually choose your u based on LIATE. But what if you were integrating a product of three functions? Say if you were integrating $x \csc(x) \log(\cos(x))$ from $0$ to $\pi/2$. What would you choose as your $u$ and $dv$?

Answer 1

Never mind acronyms: the basic idea is that you hope to end up with something easier (or at least no harder) than what you started with. If you write your integral as $\int u\; dv$, the $dv$ will need to be integrated and the $u$ differentiated. Usually integration is going to be where complications set in. So what term or product of terms can you integrate here without getting something horrible?

Well, actually, none of them is going to work. The closest is $dv = x \; dx$ which gives you $v = x^2/2$, but that gives you

$$ \frac{x^2}{2}\csc \left( x \right) \ln \left( \cos \left( x \right) \right) -\int \frac{{x}^{2}}{2} \left( -\csc \left( x \right) \cot \left( x \right) \ln \left( \cos \left( x \right) \right) -\sec(x) \right) \,{\rm d}x $$

and that seems worse than before.

In fact, in this case I'm pretty sure there is no closed-form antiderivative. Integration by parts doesn't always help!

On the other hand, if it had been $\int x \sin(x) \ln(\cos(x))\; dx$, you could take $dv = \sin(x) \ln(\cos(x))$.