The integral you want to solve is:

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

This type of integral can be solved using a trigonometric substitution. Here's a step-by-step breakdown:

### Step 1: Trigonometric substitution
We want to use a substitution that simplifies the square root expression. For integrals of the form \( \sqrt{a^2 + bx^2} \), we use the substitution:

\[
x = \frac{a}{\sqrt{b}} \tan(\theta)
\]

In your case:
- \( a^2 = 9 \) so \( a = 3 \),
- \( b = 5 \).

Thus, the substitution is:

\[
x = \frac{3}{\sqrt{5}} \tan(\theta)
\]

Now, differentiate with respect to \( x \):

\[
dx = \frac{3}{\sqrt{5}} \sec^2(\theta) d\theta
\]

### Step 2: Substituting into the integral
Substitute \( x = \frac{3}{\sqrt{5}} \tan(\theta) \) into the original integral:

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

Simplify the square root:

\[
\sqrt{9 + 5 \left(\frac{9}{5} \tan^2(\theta)\right)} = \sqrt{9(1 + \tan^2(\theta))} = \sqrt{9 \sec^2(\theta)} = 3 \sec(\theta)
\]

Now the integral becomes:

\[
\int \frac{\frac{3}{\sqrt{5}} \sec^2(\theta) d\theta}{3 \sec(\theta)}
\]

Cancel out the 3's and simplify:

\[
\int \frac{\sec^2(\theta)}{\sec(\theta)} \cdot \frac{1}{\sqrt{5}} d\theta = \frac{1}{\sqrt{5}} \int \sec(\theta) d\theta
\]

### Step 3: Solve the integral
The integral of \( \sec(\theta) \) is a standard result:

\[
\int \sec(\theta) d\theta = \ln|\sec(\theta) + \tan(\theta)| + C
\]

Thus, the integral becomes:

\[
\frac{1}{\sqrt{5}} \ln|\sec(\theta) + \tan(\theta)| + C
\]

### Step 4: Back-substitute \( \theta \)
Recall the substitution \( x = \frac{3}{\sqrt{5}} \tan(\theta) \), so:

\[
\tan(\theta) = \frac{\sqrt{5}}{3}x
\]

Using the identity \( \sec(\theta) = \sqrt{1 + \tan^2(\theta)} \), we get:

\[
\sec(\theta) = \sqrt{1 + \left(\frac{\sqrt{5}}{3}x\right)^2} = \sqrt{1 + \frac{5x^2}{9}} = \frac{\sqrt{9 + 5x^2}}{3}
\]

Now, substitute back:

\[
\frac{1}{\sqrt{5}} \ln\left|\frac{\sqrt{9 + 5x^2}}{3} + \frac{\sqrt{5}}{3}x\right| + C
\]

Simplifying:

\[
\frac{1}{\sqrt{5}} \ln\left|\frac{\sqrt{9 + 5x^2} + \sqrt{5}x}{3}\right| + C
\]

This is the final answer.

### Do you have any questions or would you like further details on any of these steps?

Here are some follow-up questions to deepen understanding:

1. What other forms of substitution are useful in integration?
2. How would the integral change if the coefficient of \(x^2\) were different?
3. Can you use partial fraction decomposition for this type of integral?
4. What is the geometric interpretation of trigonometric substitution?
5. How does hyperbolic substitution differ from trigonometric substitution?

**Tip**: When solving integrals involving square roots, always consider a trigonometric or hyperbolic substitution to simplify the expression.

Explore the step-by-step solution to the integral of dx/√(9 + 5x²) using trigonometric substitution. This problem covers key calculus concepts, including integration techniques, trigonometric substitution, and standard integral results. Ideal for undergraduate calculus students.

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