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Question
Evaluate: `int_0^(π/4) sec^4 x dx`
Solution
Let I = `int_0^(π/4) sec^4 x dx`
= `int_0^(π/4) sec^2 x . sec^2 x dx`
= `int_0^(π/4) (1 + tan^2x) sec^2 x dx`
Put tan x = t
∴ sec2 x dx = dt
Also, if x = 0, then t = 0
And if x = `π/4`, then t = 1
∴ I = `int_0^1 (1 + t^2) dt`
= `int_0^1 dt + int_0^1 t^2 dt`
= `[t]_0^1 + [t^3/3]_0^1`
= `1 + 1/3`
= `4/3`
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