_ 1 ^ 6 ([sec^ -1 ]+[cot^ -1 ])dx is equal to (where [.] denotes … - Vidyadip

MCQ

$\int_{1}^{6\pi}([sec^{-1}]+[cot^{-1}])dx$ is equal to (where $[.]$ denotes greatest integer function)

A
$12\pi-sec1$
B
$6\pi-cot1$
C
$6\pi-cot1-sec1$
✓
$6\pi-sec1$

✓

Answer

Correct option: D.

$6\pi-sec1$

d
$\int\limits_1^{6\pi } {\left( {\left[ {{{\sec }^{ - 1}}x} \right] + \left[ {{{\cot }^{ - 1}}x} \right]} \right)} dx$

$ = \int\limits_1^{6\pi } {\left[ {{{\sec }^{ - 1}}x} \right]} + \int\limits_1^{6\pi } {\left[ {{{\cot }^{ - 1}}x} \right]dx} $

$ = \int\limits_1^{\sec 1} {0.dx} + \int\limits_{\sec 1}^{6\pi } {1.dx + 0 = } 6\pi - \sec 1$

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