Integration by parts involving a function that is zero on the boundary

Statement

Statement for single application of integration by parts for functions of one variable

Let $k$ be a positive integer.

Suppose $F$ and $G$ are continuous functions whose domain of definition contains a closed interval $[a,b]$. Further, suppose both $F,G$ are differentiable on $(a,b)$, right differentiable at $a$, and left differentiable at $b$. Finally, suppose $G(a)=G(b)=0$.

$\int _{a}^{b}F(x)G'(x)\,dx=-\int _{a}^{b}F'(x)G(x)\,dx$

If we denote $F'$ by $f$ and $G'$ by $g$, then this is:

$\int _{a}^{b}F(x)g(x)\,dx=-\int _{a}^{b}f(x)G(x)\,dx$

Statement for repeated application of integration by parts for functions of one variable

Suppose $F$ and $G$ are continuous functions whose domain of definition contains a closed interval $[a,b]$. Further, suppose both $F,G$ are $k$ times differentiable on $(a,b)$, right differentiable at $a$, and left differentiable at $b$. Suppose, further, that $G(a)=G(b)=0$, the first $k-1$ right hand derivatives of $G$ at $a$ are zero, and the first $k-1$ left hand derivatives of $G$ at $b$ are zero. Then, we have:

$\int _{a}^{b}F(x)G^{(k)}(x)\,dx=(-1)^{k}\int _{a}^{b}F^{(k)}(x)G(x)\,dx$

Statement for repeated application of integration by parts for functions of one variable: compact support version

Let $k$ be a positive integer.

Suppose $F$ and $G$ are continuous functions whose domain of definition contains a closed interval $[a,b]$. Further, suppose both $F$ is $k$ times differentiable on $(a,b)$, right differentiable at $a$, and left differentiable at $b$. Suppose that $G$ is $k$ times differentiable on an open interval containing $[a,b]$, but is zero outside $[a,b]$. Then, we have:

$\int _{a}^{b}F(x)G^{(k)}(x)\,dx=(-1)^{k}\int _{a}^{b}F^{(k)}(x)G(x)\,dx$

Note that the condition of being zero outside $[a,b]$ also forces the function and all its derivatives to be zero at the endpoints $a$ and $b$

Statement for repeated application of integration by parts for functions of one variable: compact support and improper integral version

Let $k$ be a positive integer.

Suppose $F$ and $G$ are continuous functions on all of $\mathbb {R}$. Suppose both $F$ and $G$ are $k$ times differentiable everywhere, and further, suppose that $G$ has compact support, i.e., there is a closed bounded interval outside which it is zero everywhere. Then:

$\int _{-\infty }^{\infty }F(x)G^{(k)}(x)\,dx=(-1)^{k}\int _{-\infty }^{\infty }F^{(k)}(x)G(x)\,dx$