Substitution

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Theorem: Let $\mathbb{T}$ be a time scale and let $\nu \colon \mathbb{T} \rightarrow \mathbb{R}$ be strictly increasing with the property that $\tilde{\mathbb{T}}=\nu(\mathbb{T})$ is a time scale. If $f \colon \mathbb{T} \rightarrow \mathbb{R}$ is an rd-continuous function and $\nu$ is delta differentiable with rd-continuous derivative, then for all $a,b \in \mathbb{T}$, $$\displaystyle\int_a^b f(t) \nu^{\Delta}(t) \Delta t = \displaystyle\int_{\nu(a)}^{\nu(b)} (f \circ \nu^{-1})(\tau) \tilde{\Delta}\tau.$$

Proof: