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Question
Evaluate : `int_0^(pi/4) sec^4x*dx`
Solution
Let I = `int_0^(pi/4) sec^4x*dx`
= `int_0^(pi/4) sec^2x*sec^2x*dx`
= `int_0^(pi/4) (1 + tan^2x)sec^2x*dx`
Put tan x = t
∴ sec2x·dx = dt
When x = 0, t = tan 0 = 0
When x = `pi/(4), t = tan pi/(4)` = 1
∴ I = `int_0^1 (1 + t^2)*dt`
= `[t + t^3/(3)]_0^1`
= `1+ (1)/(3) - 0`
= `(4)/(3)`.
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