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Evaluation of definite integrals

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Question

$The value of \int_{0}^{100 π} (\sum_{r = 1}^{10} \tan r x) d x is equalto$

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Solution

$\begin{array}{l} Let I = \int_{0}^{100 π} \sum_{r = 1}^{10} (\tan r x) d x \\ = \int_{0}^{100 π} [\tan x + \tan 2 x + \tan 3 x + \dots \dots + \tan 10 x] d x \\ I = 100 \int_{0}^{π} [\tan x + \tan 2 x + \dots . + \tan (10 x)] d x (\sin c e \int_{0}^{n T} f (x) d x = n \int_{0}^{T} f (x) d x i f f (x) p e r i o d i s T) \\ \Rightarrow I = - 100 \int_{0}^{π} (\tan x + \tan 2 x + \dots \dots . . + \tan 10 x) d x (\sin c e \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x) \end{array}$
$t h e r e f o r e I + I = 0 \Rightarrow I = 0$

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