If you are used to the prime notation form for integration by parts, a good way to learn Leibniz form is to set up the problem in the prime form, then do the substitutions f(x) = u, g'(x)dx = dv, f'(x) = v, g(x)dx = du. Basically, the only difference is that the 'video form' uses prime notation (f'(x)), and the 'compact form' uses Leibniz notation (dy/dx). The 'compact form' is just a different way to write the form used in the videos.

I suspect however, with more practice, exposure and careful consideration, you will get it on your own. You may want to suggest to the Khan site to make a video talking about the the conversion and utility of the long form to short form notation. These articles really just serve to confirm the ubiquity of the short form notation and they may help you get you more comfortable with it: This article talks about the development of integration by parts: Same deal with this short form notation for integration by parts.

Now, since both are functions of x, for short form notation we can leave out the x. Sal writes (in the intro video)ĭ/dx = f'(x) For a moment, consider the product rule of differentiation.

Integral khan academy