Evaluate: $\int_{-10}^{20} [\cot^{-1} x] dx$. Here $[.]$ is the greatest integer function.
Step-by-Step Solution
Key Concept: General
$I = \int_{-10}^{20} [\cot^{-1} x] dx$, we know $\cot^{-1} x \in (0, \pi) \forall x \in \mathbb{R}$<br/>Thus $[\cot^{-1} x] = \begin{cases} 3, & x \in (-\infty, \cot 3) \\ 2, & x \in (\cot 3, \cot 2) \\ 1, & x \in (\cot 2, \cot 1) \\ 0, & x \in (\cot 1, \infty) \end{cases}$<br/>Hence $I = \int_{-10}^{\cot 3} 3 dx + \int_{\cot 3}^{\cot 2} 2 dx + \int_{\cot 2}^{\cot 1} 1 dx + \int_{\cot 1}^{20} 0 dx = 30 + \cot 1 + \cot 2 + \cot 3$
Correct Answer: $30 + \cot 1 + \cot 2 + \cot 3$