integrate
$$ \int \sin(x) \cos(x)\; dx $$
using $u$-substitution.
If i take $u = \sin(x)$ I get final answer to be $\sin^2(x) / 2 + c$
But If i take $u = \cos(x)$ I get final answer to be $-\cos^2(x) / 2 + c$
Are they equal? They should be, otherwise it does not make sense. But how are they equal?
The symbol $\int f(x) dx$ does not denote a function, but rather the set of all functions $F(x)$ that satisfy $F'(x)=f(x)$. Hence, any two functions in this set may differ by a constant. In your particular example, note that $\sin^2(x)/2 -(-\cos^2(x)/2)=1/2$.
Remember that for two derivable functions $\,f(x)\,,\,g(x)\,$ on some open interval $\,I\,$, we have that
$$\forall\,x\in I\;\;,\;f'(x)=g'(x)\Longleftrightarrow f(x)=g(x)+C\,\,,\,C=\,\text{ a constant}$$
Now apply the above to your problem...
