What are good applications of the generalized Integration by Parts formula? I came across the generalized Integration by Parts today:
Let $f \left({x}\right), g \left({x}\right)$ be real $n$ times differentiable functions with continuous $n$-th derivatives.
Then:
$$ \int f^{(n)}(x)g(x) dx = \sum_{j= 0}^{n-1}(-1)^j f^{(n-j-1)}(x)g^{(j)}(x)+({-1})^n \int f(x)g^{(n)}(x) dx  $$
The proof involves induction and the usual Integration by parts formula (not a surprise).
I am wondering about applications of this formula. Is there any application of the formula that cannot be obtained by a repeated use of the usual Integration by Parts formula? Or at least, that simplify a lot the use of Integration by Parts.
Note: I wrote the version for indefinite integrals. But any answer for definite integrals will equally satisfy me. 
 A: Some obvious examples:


*

*Taylor series with the integral form of the remainder. Taking the limits to be $a$ and $x$, and $f(x) = \frac{1}{n!}(x-a)^n$, $g(x) = G'(x)$, we find
$$ \begin{align}
 G(x) - G(a) &= -\sum_{j=0}^{n-1} (-1)^j \frac{n(n-1)\dotsm (n-(n-j-1)) }{n!}(x-a)^{n-(n-j-1)} G^{(j+1)}(a) + \frac{(-1)^n}{n!} \int_a^x (x-a)^n G^{(n+1)}(x) \, dx \\
G(x) &=  \sum_{k=0}^{n} \frac{(x-a)^k}{k!} G^{(k)}(a) + \frac{1}{n!}\int_a^x (a-x)^n G^{(n+1)}(x) \, dx,
\end{align}$$
which is obviously the usual Taylor series.

*Integrals of powers of logarithms. The indefinite integral/antiderivative
$$ \int x^a (\log{1/x})^b \, dx, \quad (b \in \mathbb{N}) $$
is computed using integration by parts: the best way to do this is to change variables to $x=e^{-u}$, so you have
$$ \int u^b e^{-(a+1)u} \, du, $$
and now you apply the formula with $n=b+1$, $f(u)=(-1)^b e^{-(a+1)u}/(a+1)^n$, $g(u) = u^b$ to get
$$ \int u^b e^{-(a+1)u} \, du = \sum_{k=0}^b \frac{e^{-(a+1)u}}{(a+1)^k} \frac{d^{k}}{du^{k}} (u^b) + 0, $$
since $g^{(b+1)}(x)=0$. You can then find the $k$th derivative of $u^b$, substitute back in and so on.

*A special case of this is the formula
$$ \int_0^{\infty} x^{n} e^{-ax} \, dx = \frac{n!}{a^{n+1}}, $$
probably one of the most important integrals in mathematics.

*Exponential-trigonometric integrals. Stuff of the form
$$ \int e^{ax}\cos{bx} \, dx, \quad \int e^{ax}\sin{bx} \, dx $$
can be done by using the formula with $n=2$, since we can get back to the original integrand and rearrange.


I'm sure there are plenty more.
