option pricing - Determine price of financial contract

I wonder if some one can help me with the solution to this question from Björk's "Arbitrage theory in continuous time":

At date of maturity $T_2$ the holder of a financial contract will obtain the amount: $$ \frac{1}{T_2 - T_1 } \int_{T_1}^{T_2} S(u) du $$ where $T_1$ is some time point before $T_2$. Determine the arbitrage free price of the contract at time $t$. Assume you live in a Black-Scholes world and that $t

Earlier in the book he states this theorem that I think one might use:

The arbitrage free price of a claim $\Phi(S(T))$ is given by: $$ \Pi(t,\Phi)=F(s,t) $$ where $F(\cdot,\cdot)$ is given by the formula $$ F(s,t)=e^{-r(T-t)}E_{s,t}^Q [\Phi(S(T))] $$ where the $Q$-dynamics of $S(t)$ are given by $$ dS(t)=rS(t)dt + S(t)\sigma(t,S(t))dW(t) $$

However I'm not really sure how to apply it in this case. Can anybody help me out here?