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Question
Integrate the following functions w.r.t. x : tan5x
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Solution
Let I = `int tan^5 x dx`
= `int tan^3x tan^2x dx`
= `int tan^3x (sec^2x - 1)dx`
= `int (tan^3x sec^2x - tan^3x)dx`
= `int (tan^3x sec^2x - tanx.tan^2x)dx`
= `int [tan^3x sec^2x - tanx (sec^2x - 1)]dx`
= `int (tan^3x sec^2x - tan x sec^2x + tanx)dx`
= `int[(tan^3x - tanx)sec^2x + tanx]dx`
= `int(tan^3x - tanx)sec^2x dx + inttan x dx`
= I1 + I2
In I1, put tan x = t
∴ sec2 x dx = dt
∴ I = `int (t^3 - t)dt + int tan x dx`
= `t^4/(4) - t^2/(2) + log|secx| + c`
= `tan^4x/(4) - tan^2x/(2) + log|secx| + c`.
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