You must integrate with respect to the same variable on both sides of the equation.
One integral of y' is = Integral y'(x)*dx = y(x) +c
One integral of y with respect to y is = integral y*dy = y²/2 +c
But it is of no use here, because you should integrate relatively to x , that is : Integral y*dx
Do not confuse with Integral y*dy
Since you don't know yet y(x), you cannot compute Integral y*dx. So, you need another method. The other method leads to y= c*exp(x)