Recall:
[tex]\int \frac{a}{x^{2} + a^{2}} dx = tan^{-1}(\frac{x}{a})[/tex]
[tex]\int \frac{e^{x}}{4e^{2x} + 25} dx = \int \frac{e^{x}}{4(e^{2x} + \frac{25}{4})} dx[/tex]
[tex]= \frac{1}{4}\int \frac{e^{x}}{e^{2x} + \frac{25}{4}} dx[/tex]
[tex]= \frac{1}{4}\int \frac{\frac{5}{2} \cdot \frac{2}{5} e^{x}}{e^{2x} + \frac{25}{4}} dx[/tex]
[tex]= \frac{1}{10} \int \frac{\frac{5e^{x}}{2}}{e^{2x} + \frac{25}{4}} dx[/tex]
[tex]= \frac{1}{10} tan^{-1}(\frac{2e^{x}}{5}) + C[/tex]
You can simply substitute in the bounds to find the integral from ln(5/2).
