Here are the questions I'm supposed to answer:
For what kind of integrands is integration by parts an appropriate technique to try?
What additional condition is needed for the tabular method to be used?
For the first question, I answered that the integrand must be the product of two terms. However, this is actually true for every integrand since they are just equal to $1$ times themselves.
And so, I have no idea what the best answer is to both questions.
Just to refresh your memory, the tabular method is where you have two columns, one for integration and one for differentiation, and with each row, the coefficient alternates from negative to positive. You can find examples here.
For when you should try integration by parts, I don't think that there is a clear cut answer. Rather, you should keep the technique in your toolbag, and whenever you come across an integral that you are struggling with, take it out and try to apply it.
To show why I don't think that there is a clear answer, here is one integral where integration by parts is an obvious choice: $$ \int x^2e^xdx $$ It is a product of two terms and one of the terms, $x^2$, gets simpler as we take derivatives.
Here is another integral which is easiest to solve with integration by parts $$ \int \ln(x)dx $$ Without having seen this before, I think it would be very difficult on first sight to guess that the correct method here would by integration by parts.