Integral of Lie derivative and the fundamental theorem of calculus

Question

Assume $M$ to be a smooth manifold, and denote by $j_t:M\rightarrow[0,1]\times M, x\mapsto (t,x)$ the inclusion of $M$ into $[0,1]\times M$. Let $\frac{\partial}{\partial t}$ be the canonical vector field on $[0,1]\times M$ with respect to the coordinate $t$ on the interval $[0,1]$. Then show that for any $k$-form $\beta$ on $[0,1]\times M$: $$ \int_0^1 j_t^*(\mathcal{L}_{\frac{\partial}{\partial t}}\beta)dt = j_1^*\beta - j_0^*\beta \,. $$ To me this looks a lot like a ''Lie derivative version" of the fundamental theorem of calculus, but I have not found a convincing argument that proves the above equality.