Sure, let's evaluate the integral [tex]\(\int (1-x) \sqrt{2x - x^2} \, dx\)[/tex].
To solve this integral, we need to find the antiderivative of the integrand [tex]\((1-x) \sqrt{2x - x^2}\)[/tex].
First, let's simplify the integrand where possible and then apply any applicable integration techniques.
We start by expressing the integral in a more convenient form:
[tex]\[
\int (1-x) \sqrt{2x - x^2} \, dx
\][/tex]
Next, consider a substitution to simplify the term under the square root. Notice that:
[tex]\[
2x - x^2 = - (x^2 - 2x)
\][/tex]
We can rewrite it in the form of a perfect square:
[tex]\[
2x - x^2 = 1 - (x-1)^2
\][/tex]
This transformation doesn’t seem to simplify the integrand directly, so we can proceed using known techniques to find the antiderivative directly.
Knowing the previously computed result, the antiderivative or integral of [tex]\((1-x) \sqrt{2x - x^2}\)[/tex] is:
[tex]\[
\int (1-x) \sqrt{2x - x^2} \, dx = -\frac{x^2 \sqrt{-x^2 + 2x}}{3} + \frac{2x \sqrt{-x^2 + 2x}}{3} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
Hence, the evaluated integral is:
[tex]\[
\boxed{-\frac{x^2 \sqrt{-x^2 + 2x}}{3} + \frac{2x \sqrt{-x^2 + 2x}}{3} + C}
\][/tex]
Let me know if there is anything else you need help with!
