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integrate sec x ( sec x + tan x ) dx
Take the integral: integral sec(x) (tan(x)+sec(x)) dx Expanding the integrand sec(x) (tan(x)+sec(x)) gives sec^2(x)+tan(x) sec(x): = integral (sec^2(x)+tan(x) sec(x)) dx Integrate the sum term by term: = integral sec^2(x) dx+ integral tan(x) sec(x) dx For the integrand tan(x) sec(x), substitute u = sec(x) and du = tan(x) sec(x) dx: = integral 1 du+ integral sec^2(x) dx The integral of sec^2(x) is tan(x): = integral 1 du+tan(x) The integral of 1 is u: = u+tan(x)+constant Substitute back for u = sec(x): Answer: tan(x)+sec(x)+constant