Let's break down the given problem step by step:

Problem:

The goal is to find the indefinite integral of:

$\int \frac{\sqrt{81 - x^2}}{x} \, dx$

using the substitution $x = 9\sin(\theta)$.

Step-by-step solution:

1. Substitute $x = 9\sin(\theta)$:

   • This is given as $x = 9\sin(\theta)$.
   • Differentiate this with respect to $\theta$: $dx = 9\cos(\theta) \, d\theta$

2. Simplify $\sqrt{81 - x^2}$:

   • $81 - x^2 = 81 - 81\sin^2(\theta) = 81(1 - \sin^2(\theta)) = 81\cos^2(\theta)$.
   • Therefore, $\sqrt{81 - x^2} = 9\cos(\theta)$.

3. Rewrite the integral:

   • Substituting everything into the original integral: $\int \frac{\sqrt{81 - x^2}}{x} \, dx = \int \frac{9\cos(\theta)}{9\sin(\theta)} \cdot 9\cos(\theta) \, d\theta$
   • Simplifying this: $= \int \frac{\cos(\theta)}{\sin(\theta)} \cdot 9 \cos(\theta) \, d\theta = 9 \int \frac{\cos^2(\theta)}{\sin(\theta)} \, d\theta$

4. Solve the integral:

   • Use the identity $\cos^2(\theta) = 1 - \sin^2(\theta)$ and split the integral: $9 \int \frac{\cos^2(\theta)}{\sin(\theta)} \, d\theta = 9 \int \frac{1 - \sin^2(\theta)}{\sin(\theta)} \, d\theta = 9 \int \left( \frac{1}{\sin(\theta)} - \sin(\theta) \right) \, d\theta$
   • This simplifies to: $9 \left( \ln|\sin(\theta)| + \cos(\theta) \right) + C$

5. Substitute back in terms of $x$:

   • Since $x = 9\sin(\theta)$, we know that $\sin(\theta) = \frac{x}{9}$, and $\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \frac{x^2}{81}}$.
   • Substituting these back into the solution: $9 \left( \ln\left|\frac{x}{9}\right| + \sqrt{1 - \frac{x^2}{81}} \right) + C$

Thus, the final answer is:

$9 \left( \ln\left|\frac{x}{9}\right| + \sqrt{1 - \frac{x^2}{81}} \right) + C$

Do you want more details on any specific step or have any questions?

Here are 5 related questions to deepen your understanding:

1. How do trigonometric substitutions simplify integrals involving square roots?
2. What other kinds of integrals can benefit from trigonometric substitution techniques?
3. What is the general form of the integral of $\frac{\cos^2(\theta)}{\sin(\theta)}$?
4. How would the solution change if the bounds of integration were provided?
5. Can you apply substitution if the integral contained $x^2$ instead of $x$ in the denominator?

Tip: Trigonometric substitution is especially useful when dealing with integrals involving $\sqrt{a^2 - x^2}$, $\sqrt{x^2 - a^2}$, or $\sqrt{x^2 + a^2}$.