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Question
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
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Solution
Let I = `int tan 3x tan 2x tanx dx`
Consider tan 3x = tan (2x + x)
= `(tan2x + tanx)/(1 - tan2x tanx)`
∴ tan3x (1 – tan 2x tan x) = tan 2x + tan x
∴ tan 3x – tan 3x tan 2x tan x = tan 2x + tan x
∴ tan 3x – tan 2x – tan x = tan 3x tan 2x tan x
I = `int (tan3x - tan 2x - tanx)dx`
= `int tan3xdx - int tan2x dx - int tanx dx`
= `(1)/(3)log|sec3x| - (1)/(2)log|sec2x| - log|secx| + c`.
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