31.

The function f(x) = ${\int }_0^{\mathrm{x}}\log \left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right)\mathrm{dx}$ is

• an even function

• an odd function

• a periodic function

• None of these

We know that, if f(t) is a odd function, then ${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$ is an even function. Since the function f(x) = $\log \left(\frac{1 - \mathrm{x}}{1 +\hspace{0.17em}\mathrm{x}}\right)$ is an odd function. ${\int }_0^{\mathrm{x}}\log \left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right)\mathrm{dx} \mathrm{is} \mathrm{an} \mathrm{even} \mathrm{function}.$