For $f,g\in L^1([0,1])$ with indefinite integrals $F,G$ from $0$, $\int_0^1 Fg=F(1)G(1)-\int_0^1 fG$.Suggest a slogan
Let \(f,g\in L^1([0,1])\) and put \[ F(x)\coloneqq\int_{0}^{x} f(y) \,dy, \qquad G(x)\coloneqq\int_{0}^{x} g(y) \,dy \qquad (0\leq x\leq1). \] Then \[ \int_{0}^{1} F(x) g(x) \,dx=F(1) G(1)-\int_{0}^{1} f(x) G(x) \,dx . \]