# Delta integral

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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