To find the integral [tex]\(\int \sqrt{1 - \sin(x)} \, dx\)[/tex], we want to evaluate an antiderivative for the given expression. Here's the detailed, step-by-step solution:
1. Introduce the Integral:
We start with the integral we need to solve:
[tex]\[
\int \sqrt{1 - \sin(x)} \, dx
\][/tex]
2. Consider Possible Substitutions or Transformations:
One common approach for integrals involving trigonometric functions is to use trigonometric identities or substitutions. However, in this case, a straightforward substitution does not simplify the integrand easily. For completeness, let's analyze the original integral.
3. Examine the Integral Directly:
The expression [tex]\(\sqrt{1 - \sin(x)}\)[/tex] does not have an elementary antiderivative involving basic functions and standard methods of integration, such as substitution or integration by parts, do not lead to an immediately obvious solution.
Given the steps we've followed, we may infer that the solution to the integral remains in a form that expresses the integral itself:
[tex]\[
\int \sqrt{1 - \sin(x)} \, dx
\][/tex]
Therefore, the integral of [tex]\(\sqrt{1 - \sin(x)}\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
\boxed{\int \sqrt{1 - \sin(x)} \, dx}
\][/tex]
This result indicates that the integral is best represented as an indefinite integral with the given integrand.
