A deposit of \$300 is made into a fund at time 0. The fund pays interest for the first three years at a nominal monthly rate of discount $d^{(12)}$. For $t=3$ to $t=7$, interest is credited according to the force of interest $\delta(t)=(3t+3)^{-1}$. As of time $t=7$, the accumulated value of the fund is \$574. Calculate $d^{(12)}$.
I assumed I needed to solve the equation:
$$300(1-\frac{d^{(12)}}{12})^{-12\cdot3}\cdot e^{\int_3^7 (3t+3)^{-1}dt}=574$$
for $d^{(12)}$. However, this ends up giving me a negative result, which makes no sense in the context of this problem, so clearly I'm not doing something correctly.
What is the correct equation to solve?
Your equation of value appears correct. I get $d^{(12)} \approx 0.138461$. In particular,
$$\int_3^7 \frac{dt}{3t+3} = \frac{\log 2}{3},$$ so that $$d^{(12)} = 12 \left( 1 - \left(\frac{574}{300 \cdot 2^{1/3}}\right)^{-1/36}\right).$$