How to calculate this integration about normal distribution?

Question

I want to solve a integration of the following expression: $$ \log{T} = \int_t^{t+T} \log \left[ 1 - \frac{1}{\sigma_{\tau} \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma_{\tau}^2}\right)\right] \mathrm{d}x $$ where $t$, $T$, $\sigma_{\tau}$ are constants, and $x$ is the variable.

Answer 1

HINT If you integrate by parts, letting $u = \ln(1-f(x))$ so $du = \dfrac{-f'(x)}{1-f(x)}$ and $dv = dx \implies v = x$ you get $$ \int \ln(1-f(x))dx = \int udv = uv - \int vdu = x\ln(1-f(x)) +\int\frac{xf'(x)}{1-f(x)}dx. $$ Can you do the arithmetic to simplify and integrate now? It may also help at some point to substitute $y = x/\sigma_T$ to simplify things...