TITLE: Evaluation of $\int_{0}^{10 \pi} ([\sec ^{-1}x]+[\cot^{-1} x])~\mathrm dx$
QUESTION [8 upvotes]: Find the value of the integral
$$\int_{0}^{10 \pi} (\lfloor\sec ^{-1}x\rfloor+\lfloor\cot^{-1} x\rfloor)~\mathrm dx$$
where $\lfloor . \rfloor$ denotes greatest integer function.
Could some help me with this? I cannot break greatest integer function here into different interval. Please provide some insight.

REPLY [4 votes]: $\left\lfloor \sec^{-1}x\right\rfloor=\begin{cases}
                                       0 & \text{ for } x<\sec(1)\approx1.851\\
                                       1 & \text{ for } x\ge\sec(1)
                                       \end{cases}$
$\left\lfloor\cot^{-1}x\right\rfloor=\begin{cases}
                                       1 & \text{ for } 0\le x<\cot(1)\approx0.642\\
                                       0 & \text{ for } x\ge\cot(1)
                                       \end{cases}$
So
\begin{eqnarray}
\int_0^{10\pi}\left\lfloor \sec^{-1}x\right\rfloor+\left\lfloor\cot^{-1}x\right\rfloor\,dx
&=\int_0^{\cot(1)}1+0\,dx+\int_{\sec(1)}^{10\pi}0+1\,dx\\
&=\cot(1)+10\pi-\sec(1)
\end{eqnarray}
Graph