Why do we not include $c$ in the computation of the definite integral? Why is it when evaluating the definite integral we commonly opt to omit the constant $c$
$$\int_1^2x^2 \, dx= \left.\frac{x^3}{3} \right|_1^2 =\frac{2^3}{3}-\frac{1^3}{3}=\frac{7}{3}$$
But when evaluating the indefinite integral we include it?
$$\int x^2dx = \frac{x^3}{3} + c$$
Would it not be more accurate to instead do the following?
$$\int_1^2 x^2\,dx=\left.\frac{x^3}{3}+c\right|_1^2 = \left(\frac{2^3}{3}+c\right) - \left(\frac{1^3}{3}+c\right) = \left(\frac{7}{3}+c\right)$$
Is there a historical reason to all of this?

$$
\text{Becuase } \left(\frac{2^3}{3}+c\right) - \left(\frac{1^3}{3}+c\right) = \frac{7}{3} \ne \left(\frac{7}{3}+c\right).
$$
