The integration-by-parts formula is known to be valid for functions u(x) and v(...

The integration-by-parts formula

is known to be valid for functions u(x) and v(x), which are continuous and have continuous first derivatives. However, we will assume that u, v, du/dx, and dv/dx are continuous only for a ≤ x ≤ c and c ≤ x ≤ b; we assume that all quantities may have a jump discontinuity at x = c.

*(a) Derive an expression for

(b) Show that this reduces to the integration-by-parts formula if u and v are continuous across x = c. It is not necessary for du/dx and dv/dx to be continuous at x = c