Capacitor charge over time

An electric current, I(t), flowing out of a capacitor, decays according to [1], where t is time. Find the charge, Q(t), remaining in the capacitor at time t. The initial charge is $\displaystyle Q_0$ and Q(t) is related to I(t) by [2]

1: $\displaystyle I(t) = I_{0}e^{-t}$

2: $\displaystyle Q'(t) = -I(t)$

Work so far is

$\displaystyle Q(t) = \int Q'(t)dt = -\int I(t)dt = -I_{0}\int e^{-t}dt = I_{0}e^{-t} + C = I_{0}e^{-t} + Q_0$

Though the solution seems to be

$\displaystyle Q(t)=I_{0}e^{-t}+Q_{0}-I_{0}$

What am I missing?

Re: Capacitor charge over time

$\displaystyle Q(t) = I_0e^{-t} + C$

$\displaystyle Q(0) = I_0 + C$

$\displaystyle C = Q(0) - I_0 = Q_0 - I_0$