Certainly! Let's solve the integral [tex]\(\int \sqrt{1 - \sin x} \, dx\)[/tex] step-by-step.
1. Understand the integral we need to solve:
We are asked to find the indefinite integral of [tex]\(\sqrt{1 - \sin x}\)[/tex], which is:
[tex]\[
\int \sqrt{1 - \sin x} \, dx
\][/tex]
2. Write down the integrand:
The integrand in this case is [tex]\(\sqrt{1 - \sin x}\)[/tex].
3. Identify that an elementary antiderivative for this integral doesn't exist:
The integral of [tex]\(\sqrt{1 - \sin x}\)[/tex] is a non-standard integral that cannot be expressed in terms of elementary functions such as polynomials, exponentials, logarithms, or standard trigonometric functions. Therefore, it must be represented in a special form.
4. Express the result in terms of an integral:
When an integral cannot be expressed in terms of elementary functions, it is common to write the result as an integral itself. Thus, we represent the integral of [tex]\(\sqrt{1 - \sin x}\)[/tex] by the following:
[tex]\[
\int \sqrt{1 - \sin x} \, dx = \int \sqrt{1 - \sin x} \, dx
\][/tex]
5. Conclude the solution:
Since we have determined that the integral does not have a simpler form and cannot be expressed in terms of elementary functions, the best expression we can provide is the integral itself precisely as it was originally stated.
So, the integral of [tex]\(\sqrt{1 - \sin x}\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
\int \sqrt{1 - \sin x} \, dx = \int \sqrt{1 - \sin x} \, dx
\][/tex]
This represents the most exact and simplified answer available for this integral.
