Solution
= Solution
Integrating the density, and using $\lambda_++\lambda_-=A$ and $\lambda_+\lambda_-=k_1k_2$, gives
$$
\mathbb E[\tau]
=\frac{k_1k_2}{\lambda_+-\lambda_-}
\left(\frac1{\lambda_-^2}-\frac1{\lambda_+^2}\right)
=\frac{A}{k_1k_2}
=\boxed{\frac{k_1+k'_1+k_2}{k_1k_2}}.
$$
The same result follows from <first-step analysis>: the enzyme waits on average $1/k_1$ to bind, and each complex either produces or returns to the initial enzyme state.