Working with U-substitution, I have to integrate the following.
$\int x\cos(x^2)\sin(x^2)dx$
From my understanding you can take the integral by substituting $u$ for either $\cos(x^2)$ or $\sin(x^2)$.
EG:
Solution 1:
$u = \sin(x^2)$
$\frac{1}{2}du=x\cos(x^2)dx$
$\frac{1}{2}\int udu = \frac{u^2}{4}+c = \frac{\sin^2(x^2)}{4}+c$
Solution 2:
$u = \cos(x^2)$
$-\frac{1}{2}du=x\sin(x^2)dx$
$-\frac{1}{2}\int udu = -\frac{u^2}{4}+c = -\frac{\cos^2(x^2)}{4}+c$
The two answers are shifts of each other, but are not equivalent. Is one correct and the other incorrect, or are they both valid indefinite integrals of the original function?
Since they only differ by a constant, both are correct. One might say that they both represent the same set of functions, because of the arbitrary constants.