Arccotx definite integral

Question

Need help with.the question. The solution is given but I have no idea how they have broken into intervals. Would love some help.

Answer 1

Hint: $\cot^{-1}(x)$ is a decreasing function on the entire domain, which implies that for $x \in (-\infty$; $\infty$) your resulting value will decrease from $\pi$ to $0$.

Now, remember, that $\cot^{-1}(\cot(x)) = x$ shifted by required amount of $\pi$ to be in the $(0;\pi)$. Since $0, 1, 2$ and $3$ are inside the $(0;\pi$), no shifts are needed and the result is basically $\cot^{-1}(\cot(x)) = x$ for $x \in \{0, 1, 2, 3\}$ which lets you figure out the breakpoints for the shifts and thus the points of partition of your integral.

And the last thing: $\cot(x)$ is a decreasing function, so $\cot(3) < \cot(2) < \cot(1)$