Taylor expansion of exponential of a Lie derivative

Question

In this paper on page 8 the author claims that the Taylor expansion for the expression $e^{tD_V}$ where $D_V$ is the Lie derivative with respect to a vector field $V$ (defined by $(D_VG)(x) = \frac{d}{dt}|_{t=0}V(\phi^t_V(x))$ and $\phi^t_V(x)$ is the flow of the differential equation $\dot{\psi} = V(\psi)$ looks like this

$$e^{tD_V} = I +tD_V +t^2\int_0^1(1-\theta)e^{\theta t D_V}D_V^2 d\theta $$

I can't wrap my head around the remainder term. Shouldn't it be of third oder in $D_V$ because the integrand involves the third derivative of the exponential? I was thinking that a change of variables is performed but it didn't work. A related question might be this which is concerned with the same paper but ask a different question, still.

Answer 1

Maybe I'm missing something, but it seems to me you're overthinking this. Since $$ \int_{0}^{1} d\theta (1-\theta ) \theta^{n} = \frac{1}{n+1} -\frac{1}{n+2} = \frac{n!}{(n+2)!} $$ the third term in your expression is $$ \sum_{n=0}^{\infty } \frac{t^{n+2} D_V^{n+2} }{n!} \frac{n!}{(n+2)!} = \sum_{n=2}^{\infty } \frac{t^n D_V^n }{n!} $$ exactly as it should be.