I am attempting to prove an integral that appears in the appendices of the book "Quantum Field Theory and Standard Model," specifically equation (B.25):
$$ \dfrac{1}{(2\pi)^{d}}\int\limits_{\mathbb{R}^{d}}\dfrac{k^{2a}}{(k^2-\Delta)^{b}}\text{d}^{d}k = \dfrac{i(-1)^{a-b}}{(4\pi)^{d-2}\Delta^{b-a-\frac{d}{2}}}\dfrac{\Gamma(a + \frac{d}{2})\Gamma(b - \frac{a+1}{2})}{\Gamma(b)\Gamma(\frac{d}{2})}$$
Here, $a,b\in\mathbb{N}_{0}$, where $\mathbb{N}_{0}\equiv \mathbb{N}\cup\{0\}$, and $\Delta$ is an arbitrary scalar.
The main point is that the way QFT books present things is often quite superficial, and understanding the step-by-step process can be complicated. Therefore, I would like to know how to formally prove the equation.