(1 - cos(x)) / (x - sin(x))^2
Indefinite integrals

To find the indefinite integral of \(\frac{1 - \cos(x)}{(x - \sin(x))^2}\), we can use integration techniques. However, this integral is quite complex, and direct integration is not straightforward. Let's start by attempting to simplify the expression and see if we can find a more manageable form for integration.

Consider the substitution \( u = x - \sin(x) \). Then, we have:

\[
du = \left(1 - \cos(x)\right)dx
\]

Now, we can rewrite the integral in terms of \(u\):

\[
\int \frac{1 - \cos(x)}{(x - \sin(x))^2} \, dx = \int \frac{1 - \cos(x)}{u^2} \, dx
\]

Since \( du = (1 - \cos(x)) dx \), we get \( dx = \frac{du}{1 - \cos(x)} \).

Substituting this into the integral, we get:

\[
\int \frac{1 - \cos(x)}{u^2} \cdot \frac{du}{1 - \cos(x)} = \int \frac{du}{u^2}
\]

The integral of \( \frac{1}{u^2} \) is straightforward:

\[
\int \frac{1}{u^2} \, du = -\frac{1}{u} + C
\]

Substituting back \( u = x - \sin(x) \), we get:

\[
-\frac{1}{x - \sin(x)} + C
\]

So, the indefinite integral of \(\frac{1 - \cos(x)}{(x - \sin(x))^2}\) is:

\[
\boxed{-\frac{1}{x - \sin(x)} + C}
\]

Would you like further details or have any questions?

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Tip: When dealing with complex integrals, substitution can often simplify the process by transforming the integral into a more manageable form.

Explore how to find the indefinite integral of (1 - cos(x)) / (x - sin(x))^2 with detailed step-by-step solution using trigonometric substitution method. Suitable for advanced high school students and beyond.

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