TITLE: Simple u-subsitution - Paradoxical Result
QUESTION [7 upvotes]: If I were to try and take $$\int{\mathrm{sin}(t)\mathrm{cos}(t)dt} $$ I would either take $u=\mathrm{sin}(t) $, yeilding a result of $\frac{1}{2} \mathrm{sin}^2(t) + C$, or I would take $u=\mathrm{cos}(t) $, yeilding a result of $-\frac{1}{2} \mathrm{cos}^2(t)+ C$.
These two results are not equivalent.  What just happened?

REPLY [4 votes]: The answers are not equal, but they are equivalent. Remember, as Artem said, that the two answers differ by a constant. And antiderivatives are always defined up to a constant.