Delta integral

Let $\mathbb{T}$ be a time scale. Delta integration is defined as the inverse operation of delta differentiation in the sense that if $F^{\Delta}(t)=f(t)$, then $$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)-F(s).$$

Properties of $\Delta$-integrals

Delta integral from t to sigma(t)
Delta integral is linear
Interchanging limits of delta integral
Delta integrals are additive over intervals
Integration by parts for delta integrals with sigma in integrand
Integration by parts for delta integrals with no sigma in integrand
Delta integral over degenerate interval
Modulus of delta integral
Delta integral of nonnegative function
Delta integral of delta derivative
Delta derivative of the delta integral