To solve the integral 

\[
\int \frac{dx}{\sqrt{9 + 5x^2}},
\]

we recognize that it resembles the standard form of integrals involving the square root of a sum of squares. 

### Step 1: Rewrite the Denominator

We start by rewriting the expression under the square root to fit a standard form:

\[
9 + 5x^2 = 3^2 + ( \sqrt{5}x)^2.
\]

### Step 2: Use Trigonometric Substitution

To solve this type of integral, a trigonometric substitution is useful. Since the expression is in the form \(a^2 + b^2x^2\), we use:

- Set \(x = \frac{3}{\sqrt{5}} \tan \theta\). 
- Then, \(dx = \frac{3}{\sqrt{5}} \sec^2 \theta \, d\theta\).

Now substitute \(x\) and \(dx\) into the integral:

\[
\int \frac{dx}{\sqrt{9 + 5x^2}} = \int \frac{\frac{3}{\sqrt{5}} \sec^2 \theta \, d\theta}{\sqrt{9 + 5 \left(\frac{3}{\sqrt{5}} \tan \theta\right)^2}}.
\]

Simplify the term inside the square root:

\[
9 + 5 \left(\frac{3}{\sqrt{5}} \tan \theta\right)^2 = 9 + 9 \tan^2 \theta = 9 \sec^2 \theta.
\]

Substituting this back into the integral:

\[
\int \frac{\frac{3}{\sqrt{5}} \sec^2 \theta \, d\theta}{\sqrt{9 \sec^2 \theta}} = \int \frac{\frac{3}{\sqrt{5}} \sec^2 \theta \, d\theta}{3 \sec \theta} = \int \frac{1}{\sqrt{5}} \sec \theta \, d\theta.
\]

### Step 3: Simplify and Integrate

This simplifies to:

\[
\int \frac{1}{\sqrt{5}} d\theta = \frac{1}{\sqrt{5}} \theta + C.
\]

### Step 4: Substitute Back

Since \(x = \frac{3}{\sqrt{5}} \tan \theta\), we have:

\[
\theta = \arctan\left(\frac{\sqrt{5}}{3} x\right).
\]

Substituting \(\theta\) back into the integral:

\[
\int \frac{dx}{\sqrt{9 + 5x^2}} = \frac{1}{\sqrt{5}} \arctan\left(\frac{\sqrt{5}}{3} x\right) + C.
\]

### Final Answer:

\[
\int \frac{dx}{\sqrt{9 + 5x^2}} = \frac{1}{\sqrt{5}} \arctan\left(\frac{\sqrt{5} x}{3}\right) + C.
\]

Would you like to explore any particular part of this process further? Here are some related questions you might find interesting:

1. How does trigonometric substitution work in integrals involving square roots?
2. What are the different types of substitutions for integrating similar expressions?
3. How can this approach be used to solve integrals with other constants?
4. What are the applications of these trigonometric integrals in physics?
5. How do you handle integrals involving \(\sqrt{x^2 - a^2}\) or \(\sqrt{a^2 - x^2}\)?

**Tip:** Recognizing when to use trigonometric substitution can make complex integrals much more manageable!

Learn how to solve the integral of dx/√(9+5x²) using trigonometric substitution. This solution includes rewriting the denominator, using a tangent substitution, and simplifying to get the final result. Suitable for students in Grades 11-12 and early college calculus courses.

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