Step-by-step Solution:

Rewrite the integral in terms of secant:\n\[ I = \int_{0}^{\pi/4} \sec^4 x dx \]\n\[ = \int_{0}^{\pi/4} \sec^2 x \cdot \sec^2 x dx \]\n\nUse the trigonometric identity \( \sec^2 x = 1 + \tan^2 x \):\n\[ I = \int_{0}^{\pi/4} (1 + \tan^2 x) \sec^2 x dx \]\n\nUse substitution. Let \( u = \tan x \).\nThen \( du = \sec^2 x dx \).\nChange the limits of integration:\nWhen \( x = 0 \), \( u = \tan(0) = 0 \).\nWhen \( x = \frac{\pi}{4} \), \( u = \tan(\frac{\pi}{4}) = 1 \).\n\nSubstitute these into the integral:\n\[ I = \int_{0}^{1} (1 + u^2) du \]\n\[ = \left[ u + \frac{u^3}{3} \right]_{0}^{1} \]\n\[ = \left( 1 + \frac{1}{3} \right) - (0) = \frac{4}{3} \]