My work is attached and any input would be greatly appreciated!
Thanks in advance!
algebra, algebra, algebra ...
$\displaystyle u^2+25 \ne (u-5)(u+5)$
$\displaystyle \int \frac{e^{4x}}{e^{8x} + 25} \, dx
$
$\displaystyle u =e^{4x}$
$\displaystyle du = 4e^{4x} \, dx$
$\displaystyle \frac{1}{4} \int \frac{4e^{4x}}{e^{8x} + 25} \, dx$
$\displaystyle \frac{1}{4} \int \frac{du}{u^2 + 25} \, dx$
$\displaystyle \frac{1}{20} \arctan\left(\frac{u}{5}\right) + C
$
$\displaystyle \frac{1}{20} \arctan\left(\frac{e^{4x}}{5}\right) + C$