1. **State the problem:** We need to evaluate the integral $$\int \frac{dx}{\sec 5x}$$. 2. **Rewrite the integrand:** Recall that $$\sec \theta = \frac{1}{\cos \theta}$$, so $$\frac{1}{\sec 5x} = \cos 5x$$. 3. **Simplify the integral:** The integral becomes $$\int \cos 5x \, dx$$. 4. **Use the integral formula:** The integral of $$\cos(ax)$$ with respect to $$x$$ is $$\frac{1}{a} \sin(ax) + C$$, where $$a$$ is a constant. 5. **Apply the formula:** Here, $$a = 5$$, so $$\int \cos 5x \, dx = \frac{1}{5} \sin 5x + C$$. 6. **Final answer:** $$\boxed{\frac{1}{5} \sin 5x + C}$$