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To solve the integral

\[
\int \frac{dx}{\sqrt{4 - 9x^2}},
\]

we can use a trigonometric substitution. This is because the expression under the square root, \(4 - 9x^2\), resembles the form \(a^2 - u^2\), which is suitable for substitution with \(x = \frac{a}{b} \sin \theta\).

### Steps:

1. **Rewrite the Expression:**
   Notice that:
   \[
   4 - 9x^2 = (2)^2 - (3x)^2.
   \]
   So we can set up the substitution with \(3x = 2 \sin \theta\), or equivalently, \(x = \frac{2}{3} \sin \theta\).

2. **Differentiate \(x = \frac{2}{3} \sin \theta\):**
   \[
   dx = \frac{2}{3} \cos \theta \, d\theta.
   \]

3. **Substitute into the Integral:**
   Substituting \(x = \frac{2}{3} \sin \theta\) and \(dx = \frac{2}{3} \cos \theta \, d\theta\), the integral becomes:
   \[
   \int \frac{\frac{2}{3} \cos \theta \, d\theta}{\sqrt{4 - 9 \left(\frac{2}{3} \sin \theta\right)^2}}.
   \]

4. **Simplify the Square Root:**
   Substitute \(x = \frac{2}{3} \sin \theta\) into the square root term:
   \[
   4 - 9x^2 = 4 - 4 \sin^2 \theta = 4(1 - \sin^2 \theta) = 4 \cos^2 \theta.
   \]
   So, \(\sqrt{4 - 9x^2} = \sqrt{4 \cos^2 \theta} = 2 \cos \theta\).

5. **Rewrite the Integral:**
   Substitute into the integral:
   \[
   \int \frac{\frac{2}{3} \cos \theta \, d\theta}{2 \cos \theta} = \int \frac{\frac{2}{3} \cos \theta \, d\theta}{2 \cos \theta}.
   \]
   Simplifying, we get:
   \[
   \int \frac{1}{3} \, d\theta = \frac{\theta}{3} + C.
   \]

6. **Back-Substitute \(\theta\):**
   Recall that \(x = \frac{2}{3} \sin \theta\), so \(\sin \theta = \frac{3x}{2}\). Then \(\theta = \arcsin\left(\frac{3x}{2}\right)\).

   Thus, the final answer is:
   \[
   \int \frac{dx}{\sqrt{4 - 9x^2}} = \frac{1}{3} \arcsin\left(\frac{3x}{2}\right) + C.
   \]

---

Would you like further clarification on any part of this process?

### Related Questions

1. How do you solve integrals involving other trigonometric substitutions?
2. What if the integrand involved \(\sqrt{9x^2 - 4}\) instead?
3. Can we use hyperbolic substitution for integrals of this form?
4. How would you solve integrals of the form \(\int \frac{dx}{\sqrt{a^2 + x^2}}\)?
5. How can you determine when trigonometric substitution is the best method?

#### Tip
Trigonometric substitution is especially useful when dealing with expressions under square roots that match the forms \(a^2 - x^2\), \(a^2 + x^2\), or \(x^2 - a^2\).

Evaluate the integral ∫ dx / √(4 - 9x²).

Learn how to evaluate the integral ∫ dx / √(4 - 9x²) using trigonometric substitution. This detailed solution covers step-by-step instructions, simplifying the square root and using arcsin for the final answer. Suitable for advanced high school and college students.

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