Final answer:
To find the length of the graph of f(x)=ln(4sec(x)) for 0≤x≤π/4, we need to calculate the arc length. The formula for arc length is given by: L = ∫ √(1 + (f'(x))^2) dx. In this case, we have f(x) = ln(4sec(x)). Taking the derivative f'(x), we get: f'(x) = -(4sec(x)tan(x))/(4sec(x)) = -tan(x). Substituting f'(x) into the formula for arc length and integrating from 0 to π/4, we find: L = ∫[0,π/4] √(1 + (-tan(x))^2) dx.
Step-by-step explanation:
To find the length of the graph of f(x)=ln(4sec(x)) for 0≤x≤π/4, we need to calculate the arc length. The formula for arc length is given by:
L = ∫ √(1 + (f'(x))^2) dx
In this case, we have f(x) = ln(4sec(x)). Taking the derivative f'(x), we get:
f'(x) = -(4sec(x)tan(x))/(4sec(x)) = -tan(x)
Substituting f'(x) into the formula for arc length and integrating from 0 to π/4, we have:
L = ∫[0,π/4] √(1 + (-tan(x))^2) dx
Simplifying and evaluating the integral, we find:
L = ∫[0,π/4] √(1 + tan^2(x)) dx
L = ∫[0,π/4] √(sec^2(x)) dx
L = ∫[0,π/4] sec(x) dx
To find this integral, we can use the substitution u = sec(x) + tan(x).