Given the stochastic process $S_t$ $$\frac{dS_t}{S_t} = f_tdW_t$$
Calculate $\int_0^TS_tdt$ or find its distribution
My attempt: We have $S_t = S_0\exp(-\frac{1}{2}\int_0^tf_s^2ds+\int_0^tf_sdW_s)$. Hence,
\begin{align} \int_0^TS_tdt &=TS_T - \int_0^TtdS_t \\ &=TS_T - \int_0^Ttf_tS_tdW_t \\ &=TS_0\exp(-\frac{1}{2}\int_0^Tf_t^2dt+\int_0^Tf_tdW_t) - S_0\int_0^T \left(tf_t\exp(-\frac{1}{2}\int_0^tf_s^2ds+\color{red}{\int_0^tf_sdW_s})\right)dW_t \\ \end{align}
I don't know whether it is possible to simplify the second term (the integral with the integrand containing a stochastic integral $\color{red}{\int_0^tf_sdW_s}$) such that there is no stochastic integral in integrand.
What is the distribution of the integral of a log-normal distribution?