To solve the integral
[tex]\[
\int_{-1 / 9}^0 \frac{x}{\sqrt{1-x^2}} \, dx,
\][/tex]
we'll evaluate step-by-step and find the final value.
First, we state the integrand function:
[tex]\[
f(x) = \frac{x}{\sqrt{1-x^2}}.
\][/tex]
Our goal is to integrate this function with limits from [tex]\( -\frac{1}{9} \)[/tex] to 0. This means we're looking for:
[tex]\[
\int_{-1 / 9}^0 f(x) \, dx = \int_{-1 / 9}^0 \frac{x}{\sqrt{1-x^2}} \, dx.
\][/tex]
We proceed by integrating this function over those limits. Performing this integration yields:
[tex]\[
\int_{-1 / 9}^0 \frac{x}{\sqrt{1-x^2}} \, dx = -0.006192010000093467.
\][/tex]
So, the value of the integral
[tex]\[
\int_{-1 / 9}^0 \frac{x}{\sqrt{1-x^2}} \, dx
\][/tex]
is:
[tex]\[
-0.006192010000093467.
\][/tex]
