Hadamard finite part and pricipal part of Laurent series

Question

In "Fourier analysis and its applications" by Folland, the distribution (generalized function) $X^{-k}$ is defined as $$ X^{-k}[\phi] = \frac{1}{(k-1)!} P.V. \int \frac{\phi^{(k-1)}(x)-\phi^{(k-1)}(0)}{x} dx, $$ and the formula $$ X^{-k}[\phi] = P.V.\int \frac{1}{x^k} \left[ \phi(x) - \sum_0^{k-1} \frac{\phi^{(j)}(0)}{j!} x^j \right]dx $$ is proved. Here $k$ is an integer. The integrand looks like $\phi(x)/x^k$ from which the principal part of its Laurent expansion at $x = 0$ is subtracted. Folland calls the integral as "finite part" of the integral $\int \phi(x) / x^k$. After studying this, I wanted to get some more information on the concept. So I googled of "finite part" and found some materials saying that the distribution $X^{-k}[\phi]$ actually gives "Hadamard finite part" of the integral $\int \phi(x) / x^k$. However I can't show that the formula given by Folland is actually equivalent to the Hadamard finite part. How can I show that?