The integral provided is:

$\int \frac{dx}{x^2 \sqrt{25x^2 + 9}}$

To solve this, we will use a trigonometric substitution method. The form of the integral suggests that the substitution $x = \frac{3}{5} \tan(\theta)$ will simplify the square root term. Here's the step-by-step approach:

Step 1: Substitution

Let: $x = \frac{3}{5} \tan(\theta)$ Then: $dx = \frac{3}{5} \sec^2(\theta) d\theta$

Now, substitute this into the square root term: $25x^2 + 9 = 25\left(\frac{9}{25}\tan^2(\theta)\right) + 9 = 9(\tan^2(\theta) + 1) = 9\sec^2(\theta)$ Thus: $\sqrt{25x^2 + 9} = 3\sec(\theta)$

Step 2: Substituting into the Integral

Substitute $x = \frac{3}{5} \tan(\theta)$ and $dx = \frac{3}{5} \sec^2(\theta) d\theta$ into the integral: $\int \frac{dx}{x^2 \sqrt{25x^2 + 9}} = \int \frac{\frac{3}{5} \sec^2(\theta) d\theta}{\left(\frac{9}{25} \tan^2(\theta)\right) \cdot 3\sec(\theta)}$

Simplifying the expression: $= \frac{1}{5} \int \frac{\sec^2(\theta) d\theta}{\tan^2(\theta) \sec(\theta)}$ $= \frac{1}{5} \int \frac{\sec(\theta) d\theta}{\tan^2(\theta)}$

Since $\sec(\theta) = \frac{1}{\cos(\theta)}$ and $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, we can further simplify the integral: $= \frac{1}{5} \int \frac{\cos(\theta) d\theta}{\sin^2(\theta)}$

This is a standard integral, and it can be simplified as: $\frac{1}{5} \int \csc^2(\theta) d\theta = -\frac{1}{5} \cot(\theta)$

Step 3: Back-substitution

Now, we back-substitute $\theta$ in terms of $x$. Since $x = \frac{3}{5} \tan(\theta)$, we have: $\tan(\theta) = \frac{5x}{3}$ Thus: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{3}{5x}$

Finally, the solution to the integral is: $-\frac{1}{5} \cdot \frac{3}{5x} + C = -\frac{3}{25x} + C$

Final Answer:

$\int \frac{dx}{x^2 \sqrt{25x^2 + 9}} = -\frac{3}{25x} + C$

Let me know if you need further clarification or details!

Here are 5 related questions that might interest you:

1. How do you handle integrals involving other types of trigonometric substitutions?
2. What are the key strategies for solving integrals with square roots of quadratic expressions?
3. How can we apply partial fraction decomposition in integration?
4. Can integrals involving hyperbolic functions simplify problems like this one?
5. What is the geometric interpretation of trigonometric substitution in integrals?

Tip: Always look for patterns in the form of the integral, such as $a^2 + x^2$, to choose the correct trigonometric substitution.