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$$ \DeclareMathOperator{\arccot}{arccot} \int _{ 0 }^{ a }{ \left\lfloor \arctan { x } \right\rfloor } =\int _{ 0 }^{ a }{ \left\lfloor \arccot { x } \right\rfloor } $$

The function is greatest integer function.

What is the smallest value a for which this is satisfied?

I tried a graphical approach but I'm not able to get to the solution, as domain of the both the functions is R.

Any help is appreciated!

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    Can you express $\lfloor \tan^{-1} x \rfloor$ on $[0,\rightarrow]$ as a piecewise linear function? – 2017-01-18

1 Answers 1

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Note that for $0 \le x \le a$, $0 \le \tan^{-1} x \le \pi/2$, and similarly, $0 \le \cot^{-1} x \le \pi/2$. Therefore, on this same interval, we must have $$\lfloor \tan^{-1} x \rfloor \in \{0,1\}, \quad \lfloor \cot^{-1} x \rfloor \in \{0,1\}.$$ Consequently it is not difficult to ascertain exactly when $\lfloor \tan^{-1} x \rfloor = 1$ and when $\lfloor \cot^{-1} x \rfloor = 1$. The integrals simply become the length of the intervals for which the respective functions are nonzero.

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    Can you please give a solution, I am really bad at this. I will be grateful. – 2017-01-18
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    @goku13 Why don't you show us what you tried? Asking for a complete solution without actually demonstrating any effort is, in my opinion, evidence of intellectual laziness and not worthy of serious consideration. – 2017-01-18
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    I'm sorry, that was stupid of me :p. And yeah I got the answer, is tan1+cot1. Thanks! – 2017-01-18