Answer
$$\frac{1}{\ln 5}5^{\sec \theta}+c$$
Work Step by Step
Let $u=\sec \theta$, then $du=\sec \theta \tan \theta d\theta$ and hence we get
$$\int (\sec \theta \tan \theta) 5^{\sec \theta }d\theta=\int 5^u du\\
=\frac{1}{\ln 5}5^u+c=\frac{1}{\ln 5}5^{\sec \theta}+c.$$
