Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the integral and complete the squares on the exponent and compute the Gaussian integral one interval at a time then see what simplification would come out when the number of intervals approaches infinity. Is there a more straightforward and insightful way to accomplish this? Is Fourier transform of any use due to the convolution-like form in the discretized version?