$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$
Is the taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$ at $t$ = $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\frac{d^3a}{dt^3}(t) \Delta t^4 \cdots$$?
In big O notation: Is the taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$ at $t$ = $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + O(\Delta t^2) + O(\Delta t^3) + O(\Delta t^4) \cdots$$ ?