I happened upon the following problem:
Calculate the integral $$\int_{\{x \in \mathbb{R}^n : |x|<2 \}} \frac{1}{|x|} d\lambda(x),$$ where $\lambda$ is the Lebesgue measure. The measure of a unit ball can be used as a constant.
I should already have all the necessary tools to solve this problem, but I have trouble figuring out where to start. I suppose my intuition on the behaviour of $\frac{1}{x}$ on the real numbers has become an obstacle.
Any ideas on how I could get started and make sense of it?
You will want to use generalized polar coordinates to compute this integral, a treatment of which can be found in Folland, Real Analysis, Section 2.7.
In Folland, you get the result (Corollary 2.51), that says if $f$ is a nonnegative measurable or an integrable function on $\mathbb{R}^n$ s.t. $f(x) = g(|x|)$ for some $g: (0,\infty) \to \mathbb{C}$, then $\int f dx = n\cdot m(B^n) \int_0^\infty g(r) r^{n-1} dr$, where $m(B^n)$ is the Lebesgue measure of the unit ball.