N1995 P1 Q16 [Modified]

Use the substitution \(u={{e}^{x}}\) to show that \(\displaystyle\displaystyle \displaystyle\int_{0}^{1}{\frac{{{\mathrm{e}}^{x}}-1}{{{\mathrm{e}}^{x}}+1}}\mathrm{ d}x=\displaystyle\displaystyle \displaystyle\int_{1}^{\mathrm{e}}{\frac{u-1}{u\left( u+1 \right)}}\mathrm{ d}u\). Hence, by using partial fractions, find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{1}{\frac{{{\mathrm{e}}^{x}}-1}{{{\mathrm{e}}^{x}}+1}}\mathrm{ d}x\). Suggest an alternative method.