A question of Integration by parts Consider $\int_a^b f'(x)g(x)dx$. Then the integration by parts gives $$ \int_a^b f'(x)g(x)dx = \left[ f(x)g(x) \right]_{a}^b - \int_a^b f(x) g'(x) dx.$$ In the case that $f(a), g(a), f(b), g(b)$ are not defined, can I replace the term $[f(x)g(x)]_a^b$ as $$ [f(x) g(x)]_a^b = \lim_{x \to b} f(x)g(x) - \lim_{x \to a} f(x)g(x) ? $$ Or is there any condition to justify this?

When the four values you name are not defined "prima vista" the integral in question has to be considered as an improper integral to begin with. In this case $$\int_a^b f'(x)g(x)\ dx:=\lim_{\epsilon\to0+,\ \epsilon'\to0+}\int_{a+\epsilon}^{b-\epsilon'} f'(x)g(x)\ dx$$ by definition. Now apply partial integration to the integral $\int_{a+\epsilon}^{b-\epsilon'} f'(x)g(x)\ dx$ and then proceed to the limit, if it exists.