The problem asks to find the indefinite integral of the given function using the substitution $x = 3 \tan(\theta)$. Let's proceed step by step.

We are given the integral: $\int x \sqrt{9 + x^2} \, dx$ with the substitution $x = 3 \tan(\theta)$.

Step 1: Substitution

Since $x = 3 \tan(\theta)$, we differentiate to find $dx$: $dx = 3 \sec^2(\theta) \, d\theta$

Next, substitute $x = 3 \tan(\theta)$ into $\sqrt{9 + x^2}$: $x^2 = 9 \tan^2(\theta)$ Thus: $9 + x^2 = 9 + 9 \tan^2(\theta) = 9(1 + \tan^2(\theta)) = 9 \sec^2(\theta)$ So: $\sqrt{9 + x^2} = 3 \sec(\theta)$

Step 2: Rewrite the integral in terms of $\theta$

Substitute $x = 3 \tan(\theta)$ and $dx = 3 \sec^2(\theta) \, d\theta$ into the integral: $\int x \sqrt{9 + x^2} \, dx = \int 3 \tan(\theta) \cdot 3 \sec(\theta) \cdot 3 \sec^2(\theta) \, d\theta$ This simplifies to: $27 \int \tan(\theta) \sec^3(\theta) \, d\theta$

Step 3: Solve the integral

We now need to solve: $27 \int \tan(\theta) \sec^3(\theta) \, d\theta$ We use the identity $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ and express $\sec(\theta) = \frac{1}{\cos(\theta)}$, so the integral becomes: $27 \int \frac{\sin(\theta)}{\cos^4(\theta)} \, d\theta$ We can solve this using substitution. Let $u = \cos(\theta)$, so $du = -\sin(\theta) \, d\theta$, and the integral becomes: $-27 \int \frac{du}{u^4}$ This is a standard integral: $-27 \int u^{-4} \, du = -27 \left( \frac{u^{-3}}{-3} \right) = \frac{27}{3} u^{-3} = 9 u^{-3}$ Substitute $u = \cos(\theta)$ back in: $9 \sec^3(\theta)$

Step 4: Final substitution back to $x$

Since $\sec(\theta) = \frac{\sqrt{9 + x^2}}{3}$, we have: $9 \sec^3(\theta) = 9 \left( \frac{\sqrt{9 + x^2}}{3} \right)^3 = \frac{(9 + x^2)^{3/2}}{9}$ Thus, the final answer is: $\frac{(9 + x^2)^{3/2}}{9} + C$

Conclusion:

The indefinite integral is: $\int x \sqrt{9 + x^2} \, dx = \frac{(9 + x^2)^{3/2}}{9} + C$

Do you need further details or clarifications? Here are 5 related questions to deepen understanding:

1. How do you solve integrals involving trigonometric substitutions?
2. What are the key trigonometric identities used in substitution methods?
3. How do you reverse trigonometric substitution after solving an integral?
4. Can you explain the relationship between $\tan(\theta)$ and $\sec(\theta)$ in solving integrals?
5. What are the standard forms of integrals involving $\sec^n(\theta)$?

Tip: Always remember to revert back to the original variable after solving an integral using substitution!