Question

$\int _ { - 10 } ^ { 20 } \left[ \cot ^ { - 1 } x \right] d x$, where [.] denotes greatest integer function

• A. $30 + \cot 1 + \cot 3$
• B. $30 + \cot 1 + \cot 2 + \cot 3$
• C. 8 $30 + \cot 1 + \cot 2$
• D. None of these

Answer 1

Answer: (B)

$30 + \cot 1 + \cot 2 + \cot 3$

Answer 2

Let$I = \int _ { - 10 } ^ { 20 } \left[ \cot ^ { - 1 } x \right] d x$

we know $\cot ^ { - 1 } x \in ( 0 , \pi ) \forall x \in R$

thus,

Hence,$I = \int _ { - 10 } ^ { \cot 3 } 3 d x + \int _ { \cot 3 } ^ { \cot 2 } 2 d x + \int _ { \cot 2 } ^ { \cot 1 } 1 d x + \int _ { \cot 1 } ^ { 20 } 0 d x$ $= 30 + \cot 1 + \cot 2 + \cot 3$