The problem asks us to find the correct trigonometric substitution to evaluate the integral $\int \sqrt{121 - x^2} dx$.

1. Problem Description

The problem asks us to find the correct trigonometric substitution to evaluate the integral $\int \sqrt{121 - x^2} dx$.

2. Solution Steps

We want to choose a substitution of the form $x = a \sin \theta$ such that $121 - x^2$ simplifies to a perfect square.

If we choose $x = 11 \sin \theta$, then $x^2 = 121 \sin^2 \theta$.

Substituting this into the expression under the square root gives us:

$121 - x^2 = 121 - 121 \sin^2 \theta = 121 (1 - \sin^2 \theta)$.

Using the identity $\sin^2 \theta + \cos^2 \theta = 1$, we can rewrite $1 - \sin^2 \theta$ as $\cos^2 \theta$.

So, $121 - x^2 = 121 \cos^2 \theta$.

Then, $\sqrt{121 - x^2} = \sqrt{121 \cos^2 \theta} = 11 |\cos \theta|$.

This is a reasonable simplification that leads to a solvable integral.

Now, let's check the other options.

If $x = \sqrt{11} \sin \theta$, then $x^2 = 11 \sin^2 \theta$. So, $121 - x^2 = 121 - 11 \sin^2 \theta = 11 (11 - \sin^2 \theta)$. This doesn't simplify to a perfect square.

If $x = 121 \sin \theta$, then $x^2 = 121^2 \sin^2 \theta$. So, $121 - x^2 = 121 - 121^2 \sin^2 \theta = 121 (1 - 121 \sin^2 \theta)$. This doesn't simplify to a perfect square.

If $x = 21 \cos \theta$, then $x^2 = 441 \cos^2 \theta$. So, $121 - x^2 = 121 - 441 \cos^2 \theta$. This doesn't simplify to a perfect square.

The most appropriate substitution is $x = 11 \sin \theta$.

3. Final Answer

A. $x = 11 \sin \theta$