Inverse system of a system with integral
It seems that we have the same homework. You probably are Greek.

After having the same question, I came to the conclusion that in order for this system to be invertible, you have to prove that for any given $$x_1(t),  \, x_2(t)$$ the following sentence is correct: $$ x_1(t) \ne x_2(t) \,\, {\Rightarrow} \,\, y_1(t) \ne y_2(t) $$ 

As like you proved when a math function is $"1-1"$.

For this very example, if you have the following the following two signals:

$$
\
  x_1(t) = δ(t-4)
\
$$

and

$$ x_2(t)=0 $$

it turns out that while initially $ x_1(t) \ne x_2(t) $, we have $ cos(πt/8)x_1(t) = cos(πt/8)x_2(t) $, for any $t$. This means that 

$$ \int_{t-1}^{t+1}\cos\left(\frac{\pi\tau}{8}\right)x_1(\tau)d\tau = \int_{t-1}^{t+1}\cos\left(\frac{\pi\tau}{8}\right)x_2(\tau)d\tau $$

and the system is not invertible.