Question

Evaluate ${\int }_0^{10\pi }\left(\right[{\sec }^{-1}x]+[{\cot }^{-1}x\left]\right)dx$, where $[.]$ denotes the greatest integer function.

• A. $10\pi -\cot 1$
• B. $10\pi -\sec 1$
• C. $10\pi -\tan 1$
• D. $10\pi -\sec 1-\tan 1$

Answer 1

Answer: (B)

$10\pi -\sec 1$

Here, $I={\int }_0^{10\pi }\left(\right[{\sec }^{-1}x]+[{\cot }^{-1}x\left]\right)dx$
$={\int }_0^{\sec 1}\left(\right[{\sec }^{-1}x]+[{\cot }^{-1}x\left]\right)dx+{\int }_{\sec 1}^{10\pi }\left(\right[{\sec }^{-1}x]+[{\cot }^{-1}x\left]\right)dx$
$={\int }_0^{\sec 1}(0+0)dx+{\int }_{\sec 1}^{10\pi }(1+0)dx$
$=0+(x{)}_{\sec 1}^{10\pi }=10\pi -\sec 1$