Someone who has just taken calculus and differentiated and integrated monomials $$\frac{\partial t^n}{\partial t} = nt^{n-1}$$ would say "clearly there can be no function which integrates to a constant because we get an integrative constant whenever we integrate anything."
But is there in some other sense we can define it?
Own work:
I'm thinking of for example the famous Heaviside step function which is defined as:
$$H(t)=\cases{0,\,\,t<0\\1,\,\,t\geq 0}$$
The function $t\to H(t+k)$ becomes "closer and closer" in some sense to the function $t\to f(t) = 1$ the larger $k$ gets.
In this sense, a lone Dirac impulse at the start of time $t=k\to-\infty$ would integrate to our function:
$$\int_{-\infty}^\infty\delta(t+k)dt = H(t+k)$$
Which would get closer and closer to $t\to f(t) = 1$ the larger $k$ becomes.
But how do we formulate or prove this and in which sense would it be true?
The indefinite integral of the function $ f(x)=0$ is a constant.
Because derivative of a constant is $0$
For definite integrals there are many choices.
As long as the negative and positive values of your function balance each other the integral will be $0$
For example $$\int _{-3}^3 x^5 dx =0$$
$$\int _{-2}^2 x^3 dx =0$$