To compute the integral:

\[
I = \int \sqrt{\frac{x^2 - 121}{x^2}} \, dx
\]

We begin by simplifying the integrand.

### Step 1: Simplify the integrand
The expression inside the square root can be simplified as:

\[
\sqrt{\frac{x^2 - 121}{x^2}} = \sqrt{\frac{x^2}{x^2} - \frac{121}{x^2}} = \sqrt{1 - \frac{121}{x^2}} = \sqrt{1 - \left(\frac{11}{x}\right)^2}
\]

Thus, the integral becomes:

\[
I = \int \sqrt{1 - \left(\frac{11}{x}\right)^2} \, dx
\]

### Step 2: Trigonometric substitution
This expression is similar to a standard trigonometric identity, so we apply the substitution:

\[
\frac{11}{x} = \sin\theta \quad \text{or} \quad x = \frac{11}{\sin\theta}, \quad dx = -\frac{11 \cos\theta}{\sin^2\theta} \, d\theta
\]

Substitute this into the integral:

\[
I = \int \sqrt{1 - \sin^2\theta} \cdot \left(-\frac{11 \cos\theta}{\sin^2\theta} \right) d\theta
\]

Since \(\sqrt{1 - \sin^2\theta} = \cos\theta\), the integral simplifies to:

\[
I = -11 \int \frac{\cos^2\theta}{\sin^2\theta} \, d\theta
\]

### Step 3: Simplify the trigonometric integrand
Using the identity \(\cos^2\theta = 1 - \sin^2\theta\), we rewrite the integrand as:

\[
I = -11 \int \cot^2\theta \, d\theta
\]

Now, recall that:

\[
\cot^2\theta = \csc^2\theta - 1
\]

Thus, the integral becomes:

\[
I = -11 \int (\csc^2\theta - 1) \, d\theta
\]

### Step 4: Solve the integral
Now, integrate each term:

\[
I = -11 \left( -\cot\theta - \theta \right) + C = 11 \cot\theta + 11\theta + C
\]

### Step 5: Back-substitute \(\theta\)
From the substitution \(\frac{11}{x} = \sin\theta\), we have:

\[
\sin\theta = \frac{11}{x

This problem demonstrates how to solve the integral of sqrt(x^2 - 121) / x^2 using trigonometric substitution. Follow a detailed step-by-step solution involving simplification, trigonometric identities, and integration. Suitable for undergraduate calculus students.

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