To evaluate the integral
[tex]\[ \int \sqrt{1 + \sin 2\theta} \, d\theta, \][/tex]
we start by recognizing the integrand's structure. The expression inside the square root is [tex]\(1 + \sin 2\theta\)[/tex], where [tex]\(2\theta\)[/tex] is a common trigonometric argument.
Rewrite the integral:
[tex]\[ \int \sqrt{1 + \sin 2\theta} \, d\theta. \][/tex]
At this point, let's look more closely at the expression [tex]\(1 + \sin 2\theta\)[/tex]:
1. Trigonometric Identity:
We recall the double-angle formula for sine:
[tex]\[
\sin 2\theta = 2\sin\theta \cos\theta.
\][/tex]
2. Integral Form:
Based on our expression, we need to integrate:
[tex]\[
\int \sqrt{1 + \sin 2\theta} \, d\theta.
\][/tex]
Given the result of the integral, we find it is represented in its unevaluated form:
[tex]\[ \int \sqrt{1 + \sin 2\theta} \, d\theta = \int \sqrt{\sin(2\theta) + 1} \, d\theta. \][/tex]
This representation indicates a need to use specific techniques or transformations to simplify further, which may involve advanced trigonometric identities or substitution. However, in this case, the integral is typically left in its integral form, as no simpler elementary function represents its antiderivative.
So, the evaluated form of the given integral is:
[tex]\[ \boxed{\int \sqrt{1 + \sin 2\theta} \, d\theta = \int \sqrt{\sin(2\theta) + 1} \, d\theta.} \][/tex]
