The problem statement contains a typographical error in the definition of \( I_n \). It should likely be \( I_n = \int \sec^n x \, dx \), not \( \int \sec^2 x \, dx \), as the latter does not depend on \( n \). Assuming this correction, here is the solution: ### Step 1: Establish the Reduction Formula for \( I_n = \int \sec^n x \, dx \) We start by expressing \( \sec^n x \) as \( \sec^{n-2} x \cdot \sec^2 x \) and use integration by parts: Let: - \( u = \sec^{n-2} x \) - \( dv = \sec^2 x \, dx \) Then: - \( du = (n-2) \sec^{n-3} x \cdot \sec x \tan x \, dx = (n-2) \sec^{n-2} x \tan x \, dx \) - \( v = \tan x \) Using integration by parts (\( \int u \, dv = uv - \int v \, du \)): \[ I_n = \sec^{n-2} x \cdot \tan x - \int \tan x \cdot (n-2) \sec^{n-2} x \tan x \, dx \] \[ I_n = \sec^{n-2} x \tan x - (n-2) \int \sec^{n-2} x \tan^2 x \, dx \] Substitute