Find : \[\int \frac{\mathrm{e}^{\mathrm{x}}}{\sqrt{\mathrm{e}^{2 \mathrm{x}}-4 \mathrm{e}^{\mathrm{x}}-5}} d \mathrm{x} \]

Let $\displaystyle \mathrm{e}^{\mathrm{x}}=\mathrm{t}$. Then $\displaystyle \mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$ Given integral becomes\begin{aligned} & \int \frac{d \mathrm{t}}{\sqrt{\mathrm{t}^{2}-4 \mathrm{t}-5}} = & \int \frac{d \mathrm{t}}{\sqrt{(\mathrm{t}-2)^{2}-3^{2}}} \end{aligned}$\displaystyle =\log \left|(\mathrm{t}-2)+\sqrt{\mathrm{t}^{2}-4 \mathrm{t}-5}\right|+\mathrm{C}$=\log \left|\mathrm{e}^{\mathrm{X}}-2+\sqrt{\mathrm{e}^{2 \mathrm{x}}-4 \mathrm{e}^{\mathrm{x}}-5}\right|+\mathrm{C} $$