Question Detail
Question
Given that ∫0θ 5 tan x sec4 x dx = 15/4, where 0 < θ < π/2, find the exact value of θ.
Correct Answer
Option A is the correct answer.
Detailed Explanation
Rewrite the integrand: 5 tan x sec4 x = 5 sec³ x (sec x tan x) dx.Let u = sec x, then du = sec x tan x dx.Substitute into the integral: ∫ 5 u³ du = 5 (u⁴/4) = (5/4)sec⁴ x.Now evaluate the definite integral from 0 to θ:[(5/4)sec⁴ x]0θ = (5/4)[sec⁴ θ - sec⁴(0)].Since sec(0) = 1, sec⁴(0) = 1.So, (5/4)[sec⁴ θ - 1].We are given that this equals 15/4:(5/4)[sec⁴ θ - 1] = 15/4.Divide by 5/4:sec⁴ θ - 1 = 3.sec⁴ θ = 4.Taking the fourth root, sec θ = ±&sqrt;2.Since 0 < θ < π/2, sec θ must be positive. So, sec θ = &sqrt;2.This means cos θ = 1/&sqrt;2.The angle θ in the range (0, π/2) for which cos θ = 1/&sqrt;2 is θ = π/4.
Hint
Use u-substitution with u = sec x. Remember to adjust the limits of integration or evaluate the indefinite integral and then apply the original limits. Simplify the integrand first if needed.