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Question
Evaluate the following:
`int sec^3x.dx`
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Solution
Let I = `int sec^3x.dx`
= `int sec x sec^2x.dx`
= `sec x int sec^2x.dx - int[d/dx(secx) int sec^2x.dx].dx`
= `secx tanx- int (secx tanx)(tanx).dx`
= `secx tanx - int secx tan^2x.dx`
= `secx tanx - int secx (sec^2x - 1).dx`
= `secx tanx - int sec^3x.dx + int secx.dx`
∴ I = sec x tan x – I + log |sec x + tanx|
∴ 2I = sec x tan x + log |sec x + tan x|
∴ I = `(1)/(2)[secx tanx + log |secx + tan|] + c`.
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