Integrate only using differentiation with respect to parameter:
$$I=\int_0^\infty \frac{1}{(x^2+p)^{n+1}}dx, \quad n \geq 0, \quad p \geq 1$$
No complex methods allowed. This is a rather useful integral to know. By differentiation with respect to parameter, I mean writing something like
$$I(\alpha)=\int_0^\infty f(\alpha,x) dx,\ \to I'(\alpha)=\int_0^\infty \partial_\alpha f(\alpha,x) dx=\frac{d}{d\alpha}\int_0^\infty f(\alpha,x) dx.$$
These integrals are mostly taken from Bulgarian math team questions. Thanks again for the assistance and your time and help!
Hint: Let $I(p)=\displaystyle\int_0^\infty\frac{dx}{x^2+p}$ , and then try to express $\displaystyle\int_0^\infty\frac{dx}{(x^2+p)^{n+1}}$ in terms of $I^{(n)}(p)=$ $=(-1)^n\dfrac{(2n-1)!!}{2^{n+1}p^n\sqrt p}\cdot\pi$.