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The integral given is:

\[
\int \frac{x^3}{\sqrt{x^2 + 9}} \, dx
\]

The problem asks to use an appropriate trigonometric substitution to transform this into another form. Here's how we solve it step by step:

### Step 1: Choose the trigonometric substitution
We have the term \(\sqrt{x^2 + 9}\), so we can use the substitution:

\[
x = 3\tan\theta
\]

This substitution works because:

\[
x^2 + 9 = 9\tan^2\theta + 9 = 9(\tan^2\theta + 1) = 9\sec^2\theta
\]

Thus, \(\sqrt{x^2 + 9} = 3\sec\theta\).

### Step 2: Differentiate \(x = 3\tan\theta\)
Taking the derivative of both sides:

\[
dx = 3\sec^2\theta \, d\theta
\]

### Step 3: Substitute into the integral
Now substitute \(x = 3\tan\theta\) and \(dx = 3\sec^2\theta \, d\theta\) into the integral:

\[
\int \frac{(3\tan\theta)^3}{3\sec\theta} \cdot 3\sec^2\theta \, d\theta
\]

This simplifies as follows:

\[
\int \frac{27\tan^3\theta}{\sec\theta} \cdot 3\sec^2\theta \, d\theta = 27 \int \tan^3\theta \sec\theta \, d\theta
\]

### Step 4: Conclusion
Thus, the integral becomes:

\[
27 \int \tan^3\theta \sec^2\theta \, d\theta
\]

This matches option **A** in the choices.

### Answer: The correct option is **A**.

---

Do you have any questions or would you like further details on any of the steps? Here are five related questions you might find helpful:

1. What are the common types of trigonometric substitutions used in integrals?
2. How do you simplify integrals involving powers of trigonometric functions like \(\tan^3\theta\)?
3. Why is the substitution \(x = 3\tan\theta\) appropriate in this case?
4. What are other methods to solve integrals involving square roots and polynomials?
5. How does differentiation apply to trigonometric functions like \(\sec^2\theta\) and \(\tan\theta\)?

**Tip:** When working with trigonometric substitutions, remember the identity \( \tan^2\theta + 1 = \sec^2\theta \). This identity is crucial in simplifying the square root terms.

Using the appropriate trigonometric substitution, simplify the integral ∫ (x^3 / √(x^2 + 9)) dx.

Learn how to use trigonometric substitution to solve the integral ∫ x^3 / √(x^2 + 9) dx, using the substitution x = 3tan(θ). This detailed step-by-step guide covers key calculus concepts and is suitable for undergraduate students.

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