Research proof index

An integration-by-parts identity for real- or complex-valued $f$. If $a\le b$, $f$ is differentiable on $[a,b]$, and $f'$ is integrable there, then for every constant $c$, $\int_a^b(t+c)f'(t)\,dt=(b+c)f(b)-(a+c)f(a)-\int_a^b f(t)\,dt.$

For $0\le a\le b$ and a differentiable function $f$ whose derivative is continuous on $[a,b]$, $\int_a^b f'(t)(\lfloor t\rfloor+1)\,dt=(b+\tfrac12)f(b)-(a+\tfrac12)f(a)-\int_a^b f(t)\,dt-\int_a^b f'(t)B_1(t)\,dt,$ where $B_1(t)=t-\lfloor t\rfloor-\tfrac12$.

A first-order Euler–Maclaurin formula. For $0\le a\le b$ and a differentiable function $f$ whose derivative is continuous on $[a,b]$, the sum over integers $\lfloor a\rfloor<k\le\lfloor b\rfloor$ equals $f(a)B_1(a)-f(b)B_1(b)+\int_a^b f(t)\,dt+\int_a^b f'(t)B_1(t)\,dt,$ where $B_1(t)=t-\lfloor t\rfloor-\tfrac12$.