Consider an integral:
$$I^{ij}=\int\frac{d^d\textbf{p}}{(2\pi)^d}p^i p^j f(\textbf{p}^2).$$ How can we show that this is equal to: $$I^{ij}=\frac{\delta^{ij}}{d}\int\frac{d^d\textbf{p}}{(2\pi)^d}\textbf{p}^2 f(\textbf{p}^2).$$
My attempt is to rewrite like $$I^{12}=\int^{\infty}_{-\infty}dp_1 dp_2 p_1 p_2\int\frac{d^{d-2}\textbf{p}_q}{(2\pi)^d}f(p^2_1+p^2_2+\textbf{p}^2_q)$$ where we separated the rotationally invariant parts, and probably we need to do a clever change of variable transform somehow.
