\(\int \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=2 \tan ^{-1}(\mathrm{f}(x))+\mathrm{c}\) where \(x\gt0\) and c is a constant of integration, then \(\mathrm{f}(x)\) is

Let \(I=\int \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}\)
Let \(\sqrt{\mathrm{e}^x-1}=\mathrm{t}\)
\(\therefore \mathrm{e}^x-1=\mathrm{t}^2 \)
\( \therefore \mathrm{e}^x=\mathrm{t}^2+1 \)
\( \therefore \mathrm{e}^x \mathrm{~d} x=2 \mathrm{t} \mathrm{dt} \)
\( \therefore \mathrm{~d} x=\frac{2 \mathrm{t}}{\mathrm{e}^x} \mathrm{dt}=\frac{2 \mathrm{t}}{\mathrm{t}^2+1} \mathrm{dt} \)
\( \begin{aligned} \therefore \mathrm{I} \end{aligned}=\int \frac{1}{\mathrm{t}} \times \frac{2 \mathrm{t}}{\mathrm{t}^2+1} \mathrm{dt} \)
\( =2 \int \frac{1}{\mathrm{t}^2+1} \mathrm{dt} \)
\( =2 \tan ^{-1}(\mathrm{t})+\mathrm{c} \)
\( =2 \tan ^{-1}\left(\sqrt{\mathrm{e}^x-1}\right)+\mathrm{c} \)
\( \therefore \mathrm{f}(x)=\sqrt{\mathrm{e}^x-1}\)
\(\ldots[\) from (i)]