Question

Integrate log tanx Lim 0 to pie/2

Answer 1

 Best Answer

Note : ∫ [ 0, a ] ƒ(x) dx = ∫ [ 0, a ] ƒ(a-x) dx ....... (1) 
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I = ∫ [0,π/2] ln ( tan x ) dx .................................. (2) 

From (1), replacing (x) by (π/2 - x), 

I = ∫ [0,π/2] ln [ tan (π/2 - x) ] dx 

..= ∫ [0,π/2] ln ( cot x ) dx 

..= ∫ [0,π/2] ln [ ( tan x )ֿ¹ ] dx 

..= - { ∫ [0,π/2] ln ( tan x ) dx } 

..= -I .............. from (2) 

Thus, 

I = -I ∴ 2·I = 0 ∴ I = 0 .................. Ans. 
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