Discuss the role of constant of integration in indefinite integrals, why we don't involve it in definite integrals? (Be precise)

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Let's take Newton-Leibniz formula:
`int_a^bf(x)dx=F(b)-F(a)`
where `F(x)=intf(x)dx`.
So if we were to to calculate indefinite integral of e.g.`f(x)=x`
we would get
`F(x)=int xdx=x^2/2+C`.
Let's now calculate definite integral:
`int_a^b x dx=b^2/2+C-a^2/2-C=b^2/2-a^2/2`
So we see that constants cancel each other so there is no need to write them at all. Also `C` can be any constant so we can choose it to be 0.

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