Indefinite integral of a constant

Question

$$\int k \,\mathrm{d}x = k+c $$

Why is it that the upon integral is equal to $k+c$? Shouldn't it be $kx+c$? Since it's a constant. When you integrate constants you get $ax$ or $kx$?

Answer 1

The integral calculates an expression that when it gets differentiated, it gives you the expression that was inside the integral. This means that :

$$\int kdx=kx+c$$

if $k \space \text{constant} \space \in \mathbb R$.

Double checking by what I mentioned above :

$$(kx+c)'=(kx)' + c' = k$$

which yields you the initial expression.