Definition

Suppose $f, g : [a, b] \to R$ is continuous on a $[a, b]$ , differentiable on a $(a, b)$ , and $f^{'}, g^{'} \in R [a, b]$ . Then, $\int_{a}^{b} f^{'} (x) g (x) d x = [f (x) g (x)]_{a}^{b} - \int_{a}^{b} f (x) g^{'} (x) d x = f (b) g (b) - f (a) g (a) - \int_{a}^{b} f (x) g^{'} (x) d x$