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The value of the integral ∫[tan^–1(cotx) + cot^–1(tanx)]dx for x ∈ [0, π/2] is ……………

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The value of the integral ∫[tan–1(cotx) + cot–1(tanx)]dx for x ∈ [0, π/2] is …………… 

(a) (π/4) 

(b) π 

(c) (π2/4) 

(d) (π2/2)

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Best answer

The correct option (c) (π2/4)

Explanation:

I = (π/2)∫0 [tan–1(cotx) + cot–1(tan x)]dx 

= (π/2)∫0 [tan–1{cot[(π/2) – x]} + cot–1{tan[(π/2) – x]}] dx 

= (π/2)∫0 tan–1 (tan x) + cot–1(cotx)] dx 

= (π/2)∫02xdx = (π2/4).

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