# Probability of a machine working at certain time

This question arises while I am learning Continuous Time Markov Chain :

A machine is working for an exponential time with rate $$\mu$$ before breaking down. The repair time of the machine is exponentially distributed with rate $$\lambda$$. Find the probability that the machine is working at a certain time $$T$$.

Note : The pdf of an exponentially distributed random variable with rate $$\lambda$$ is $$\lambda e^{-\lambda x}$$

My idea is to conditioning on what time the machine starts breaking down and all I get is $$P[\text{Machine working at time T}] = \int_0^T P[\text{Machine working at time T} | \text{Machine is breaking down at x}] \mu e^{-\mu x} ~ dx$$ and I couldn't proceed any further.

Any ideas on how to solve it? Thanks!