Doubt in proof of integrating by part

Question

In the proof of integration by parts given in the photo below I see f(X)•g(x) get differntiated and I see the dx below them move to the RHS, so I left hand side we should have d(f(X)•g(X)) right? I am not able to understand how did dx come in the LHS

Answer 1

This proof keeps repeating the same typo. When they write $\int fgdx$, it should be $fg$. Or if we work with definite integrals, $\int_a^b(fg^\prime+gf^\prime)dx=[fg]_a^b=f(b)g(b)-f(a)g(a)$. However, the later special case $g=\int vdx$ has been handled correctly.