Is the following system time-invariant or time-varying?
Homework, I guess => only guidance is available.

No doubt, this is a linear system. Calculate the response y(t) when x(t) is the impulse function del(t). With that input the integration is trivial. 

Then calculate the response with delayed impulse input del(t-A). The integration is still trivial. Compare the responses. The system is time-varying if there's more difference between the responses than the used time delay A. If you find even one A which causes more difference than just the time delay =A, the system is proven to be time variant. To prove the system to be time independent, you must show that every A, positive or negative, causes amount A delayed response.

BTW. You have some dim areas to be cleared at first. As commented, factor exp(-t) is NOT a time delay. You are now in time domain. Integrations stay non-trivial until you understand the sampling property of the impulse function.