Integrating Using Integration by Parts

The product rule says (uv)' = u'v + uv'. Integrate both sides and rearrange, and you get integral of u dv = uv - integral of v du. That's integration by parts. It matters because plenty of integrands are products of two different kinds of functions, like x times e^x, where no substitution cleans things up. Parts shifts the difficulty: you trade your original integral for a new one, integral of v du, and the whole trick only pays off if that new integral is easier than the one you started with.

The hard part is choosing u and dv wisely. Pick u to be the factor that gets simpler when you differentiate it, and dv to be the factor you can actually antidifferentiate. A rough priority order: let u be a log or inverse trig function first, then a polynomial, before ever letting u be exponential or trig. For x e^x, let u = x (its derivative, 1, is simpler) and dv = e^x dx (easy to integrate). Going the other way, with u = e^x and dv = x dx, makes the new integral involve x² e^x, which is worse, not better. For x ln(x), let u = ln(x) since its derivative 1/x is cleaner algebraically, and dv = x dx.

Sometimes the new integral is still hard, or it even loops back to a multiple of the original integral. When that happens, you either apply parts again to the new integral, or solve algebraically: if the original integral reappears, treat it as an unknown, move all copies to one side, and divide it out. This connects integration by parts to the rest of the toolkit in the unit, since it's often the last trick you reach for once substitution, partial fractions, and trig integrals have failed.