How to integrate functions involving periodic bernoulli function like $$\int_1^n\frac{-6}{x^4}P_{4}(x)dx$$ where $P_{4}(x) = B_{4}(x - \lfloor x \rfloor) \text{ is the 4th order periodic bernoulli function}$ ?
Here's my attempt, please verify if the following approach is the correct way to integrate.
Let $u =P_{4}(x), dv = \frac{-6}{x^4}dx$
Then $v = \int{\frac{-6}{x^4}dx} = \frac{2}{x^3}$
$$\int{udv}= uv - \int{vdu}$$ $$\int udv= \frac{2}{x^3}P_{4}(x) - \int{{4P_3(x)}\left(\frac{2}{x^3}\right)}dx $$
$$\int_1^n\frac{-6}{x^4}P_{4}(x)dx = \left|\frac{2}{x^3}P_{4}(x) - \int{{4P_3(x)}(\frac{2}{x^3})}dx\right|_1^n\dots\text{ and so forth.}$$