i am using the above mental model in order to make sense of the calculation of the integral. My mental model could possibly be erroneous. i want to prove this: $$ \int_0^T\,x(t)\,dt = \int_a^{a+T}\,x(t)\,dt\,\\where\,x(t)\,is\,some\,periodic\,function,unknown $$
as a hint it is given that: $\int_a^0\,x(t)\,dt = \int_{a+T}^T\,x(t)\,dt$
i do not understand this specific part of the proof which is the first step of the proof(!),that is how it starts proving it: $ \int_a^{a+T}\,x(t)\,dt = \int_{a+T}^T\,x(t)\,dt+\int_0^{a+T}x(t)\,dt$
I tried to reason this first step by using my mental model but to no avail. Maybe the book has skipped a step which i do not see. Can you help please? (what comes after that first step is easily understood)
$$\begin{aligned}\int_a^{a+T}x(t)dt&=\int_0^{a+T}x(t)dt-\int_0^a x(t)dt \\&= \int_0^{a+T}x(t)dt+\int_a^0 x(t)dt \\&= \int_0^{a+T}x(t)dt+\int_{a+T}^Tx(t)dt\end{aligned}$$ where the last step uses the hint.