# Closed form solution for differential equation

I would like to solve the following equation:

$$\bar{t} = \int_0^{\infty} t F(t) \, dt = \int_0^{\infty} e^{ -4 ms \int_0^t p(t') \, dt' } \, dt$$

Just focusing on the exponential section:

$$p(t')dt'$$

and changing the notation slightly to substitute $$t'$$ with $$u$$.

I have worked out a few of the steps (see below) but I am not sure if the result has a closed form solution? The following are constants $$m$$, $$s$$, $$N$$.

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