Consider the following indefinite integral.

Question

I= ∫ tan(x)sec^2(x) dx
Part 1 of the question was to evaluate I using substitution, u=sec(x)
my answer: (sec^2(x))/2 +C
Part 2: evaluate I using substitution, u=tan(x)
my answer: tan^2(x)/2 + C

Part 3 is to explain why the solutions to parts 1 and 2 are seemingly different.
I would like to know whether my answers for part 1 and 2 are correct and would like help with explaining in part 3.

Answer 1

it's that pesky C
sec^2 = tan^2 + 1