> [!NOTE] Theorem (Integration by Parts) > Let $f,g$ be [[Real Function|real functions]] on a [[Closed Real Interval|closed real interval]] $[a,b]$ that are [[Fréchet Differentiation|differentiable]] on an [[Open Real Interval|open real interval]] containing $[a,b]$ such that their [[Derivative of Real Function|derivatives]] $f',g'$ are [[Riemann integration|Darboux integrable]] on $[a,b].$ Then $\int_{a}^{b} f(x)g'(x) \, dx = f(b)g(b)-f(a)g(a) - \int_{a}^{b} f'(x)g(x) \, dx $ **Proof**: By [[Derivative of Product of Differentiable Real Functions]], $(fg)'=f'g+fg'$ and each term is integrable. So $ \int_a^b f(x) g^{\prime}(x) d x+\int_a^b f^{\prime}(x) g(x) d x=\int_a^b(f g)^{\prime}(x) d x=f(b) g(b)-f(a) g(a) . $ # Applications See https://people.math.harvard.edu/~knill/teaching/math1a_2012/handouts/39-parts.pdf