Evaluate the following:

i)$\int \frac{3{e}^{x}}{{e}^{x}-1}dx$

ii) ${\int}_{0}^{1}X\mathrm{cos}(\pi X)dX$

iii)$\underset{x\to 1}{lim}\frac{x-1}{\sqrt{x}-1}$

iv)$\underset{x\to \mathrm{\infty}}{lim}\frac{x}{\mathrm{ln}(1+3{e}^{x})}$

i)$\int \frac{3{e}^{x}}{{e}^{x}-1}dx$

ii) ${\int}_{0}^{1}X\mathrm{cos}(\pi X)dX$

iii)$\underset{x\to 1}{lim}\frac{x-1}{\sqrt{x}-1}$

iv)$\underset{x\to \mathrm{\infty}}{lim}\frac{x}{\mathrm{ln}(1+3{e}^{x})}$