Instead of using the formula \int f(x)g'(x)dx = f(x)g(x) - \int f'(x)g(x)dx, would \int f'(x)g(x)dx = f(x)g(x) - \int f(x)g'(x)dx work with solving integration by parts problems? Based on the derivation of this function from the product rule in differentiation, it seems like it would work (since they are equivalent) but the numbers seem to get much uglier (at least, from the problems that I have attempted this secondary formula with).
I tried to solve $\int x\cdot \sin (2x)dx$ in this way, and easily obtained the answer using the regular formula, but the numbers become different as you solve it with the secondary formula.
Thank you!