How can i avoid mistakes when choosing u and dv in integration by parts?

[SavannahB]

I'm working through some integration by parts problems and I keep getting stuck when the integrand is a product like x times a trigonometric function. I tried to solve ∫ x sin(x) dx and my final answer had the sign wrong on the cosine term. I think I'm making a mistake when I choose my u and dv, or maybe when I do the subsequent differentiation and integration.

[Chloe.M]

I tend to start with u = x and dv = sin x dx. Then du = dx and v = -cos x. So ∫ x sin x dx = -x cos x + ∫ cos x dx = -x cos x + sin x + C. The sign on the cos term comes from differentiating cos.

[Edward36]

I've messed up that sign plenty of times too. I double-check by differentiating the result: d/dx(-x cos x + sin x) = -cos x + x sin x + cos x = x sin x. It feels obvious after you write it, but I still worry I mixed pieces at first when choosing u and dv.

[Samuel.W]

Could the real problem be that you expect one pass to finish it? with u = x and dv = sin x you still end up with -x cos x plus sin x, so you need that extra integral piece. If you misunderstand which piece is being integrated, the signs get tangled.

[Sofia_G]

I remember the first time I solved this I had a pile of scratch paper and a timer, and the signs kept tripping me up. I tried a few wrong swaps of u and dv, and I abandoned some attempts because the algebra got too messy. I keep checking by differentiating the result now, so at least I know when it went right.