First slide

Evaluation of definite integrals

Question

$Evaluate \int_{0}^{10 π} [\tan^{- 1} x] d x, where [\cdot] represents greatest integer function.$

Moderate

A

$10 π$
B

$10 π - \tan 1$
C

$10 π + \tan 1$
D

$10 π + \tan 2$

Solution

$Here, y = \tan^{- 1} x is a monotonic function.$
$We have \{\begin{cases} 0 < \tan^{- 1} x < 1; when 0 < x < \tan 1 \\ 1 < \tan^{- 1} x < \frac{π}{2}; when \tan 1 < x < 10 π \end{cases}$
$∴ I = \int_{0}^{10 π} [\tan^{- 1} x] d x = \int_{0}^{\tan 1} 0 d x + \int_{\tan 1}^{10 π} 1 d x$
$= 10 π - \tan 1$

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