To evaluate the given integral:
$\int_0^{\pi/4} \sqrt{\cos 2t} \, dt$ we proceed as follows:

This integral does not have a standard elementary solution, but we can express it in terms of special functions or approximate it numerically. However, if there are additional simplifications available based on prior steps (which are not shown in the image), they may help in evaluating this integral.

Do you want a numerical approximation or a solution in terms of known functions? Let me know how detailed you want the explanation.

Related Questions:

1. How do you derive the arc length formula from first principles?
2. What substitution methods can be used for integrals involving square roots of trigonometric functions?
3. What are elliptic integrals, and how do they relate to non-elementary integrals?
4. Can numerical integration methods approximate $\int_0^{\pi/4} \sqrt{\cos 2t} \, dt$?
5. How does the cosine function affect the behavior of the integral in different intervals?

Tip: When dealing with integrals of the form $\int \sqrt{\cos 2t} dt$, consider trigonometric identities or numerical approximation methods.