Let $f,g:[a,b]\to\mathbb{R}$ be continuously differentiable (i.e., $f,g\in C^1([a,b])$). Then $f'$, $g'$, $fg$ are continuous and hence Riemann integrable .

Corollary (integration by parts): $ \int_a^b f(x),g’(x),dx

f(b)g(b)-f(a)g(a)-\int_a^b f’(x),g(x),dx. $

Integration by parts is a fundamental transformation tool in analysis, especially for estimating integrals and manipulating Fourier-type expressions.

Connection to parent theorem: Apply the product rule $(fg)'=f'g+fg'$ and integrate both sides: $\int_a^b (fg)'(x)\,dx = \int_a^b f'(x)g(x)\,dx+\int_a^b f(x)g'(x)\,dx.$ By the fundamental theorem of calculus , $\int_a^b (fg)' = (fg)(b)-(fg)(a)$, giving the formula.