1. How does it possible for∫sin(x)cos(x) dx to have two different solutions, namely (1/2)*(sin^{2}x)+c and -(1/2)*(cos^{2}x)+c, so that would bring to (rediculous) conclusion that sin^{2}x+cos^{2}x=0?

2. Integration by substitution makes use of differentials (not simply derivatives), thus is it purely anti-derivative?

Thank you in advance