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We can solve this integral using a substitution method.Let (u = \sec x). Then (du = \sec x \tan x\,dx).The integral can be rewritten as (\int \sec x (\sec x \tan x)\,dx = \int u\,du).This evaluates to (\frac{u^2}{2} + C).Substituting back (u = \sec x), we get (\frac{\sec^2 x}{2} + C).Alternatively, let (u = \tan x). Then (du = \sec^2 x\,dx).The integral becomes (\int u\,du = \frac{u^2}{2} + C = \frac{\tan^2 x}{2} + C).Since (\sec^2 x = 1 + \tan^2 x), we have (\frac{1}{2}\sec^2 x = \frac{1}{2}(1+\tan^2 x) = \frac{1}{2} + \frac{1}{2}\tan^2 x). Both (\frac{1}{2}\sec^2 x) and (\frac{1}{2}\tan^2 x) are valid antiderivatives, differing by a constant. Given the options, (\frac{1}{2}\sec^2 x) is explicitly listed.