https://file.aiquickdraw.com/mathbot/file/17256036972061ehi8aoq.jpg

The integral shown in the image is:

\[
\int \frac{dx}{(81x^2 + 64)^2}
\]

This type of integral can be solved using a trigonometric substitution or other integration techniques. First, let's simplify and work towards a solution.

The expression inside the denominator is of the form \(a^2x^2 + b^2\), which suggests a possible trigonometric substitution. Let me go through the steps to solve this integral.

### Step-by-step Solution:

1. **Simplify the expression**: Notice that the denominator has the form \( (81x^2 + 64)^2 \). Rewriting the constants as squares, we get:

   \[
   \int \frac{dx}{(9^2 x^2 + 8^2)^2}
   \]

2. **Substitution**: For this type of expression, we can use the substitution:

   \[
   x = \frac{8}{9} \tan(\theta)
   \]
   This implies \( dx = \frac{8}{9} \sec^2(\theta) d\theta \).

3. **Change of variables**: Substitute \( x = \frac{8}{9} \tan(\theta) \) into the integral and simplify the resulting expression to find the solution.

Let me compute the final solution for you.The solution to the integral

\[
\int \frac{dx}{(81x^2 + 64)^2}
\]

is:

\[
\frac{x}{10368x^2 + 8192} + \frac{\tan^{-1}\left(\frac{9x}{8}\right)}{9216} + C
\]

where \( C \) is the constant of integration.

Would you like further details on this solution, or any clarification? Here are some related questions to deepen your understanding:

1. How does the substitution method work for integrals involving quadratic terms?
2. Why is trigonometric substitution useful in this context?
3. Can this result be verified by differentiation?
4. How does the \(\tan^{-1}(x)\) function arise in integration of rational functions?
5. What other types of substitutions are common for integrals of this form?

**Tip**: When working with complex integrals, recognizing standard forms (like \(\int \frac{dx}{a^2 + x^2}\)) can simplify the solution process.

Evaluate the integral of 1 / (81x^2 + 64)^2 with respect to x.

Learn how to evaluate the integral of 1 / (81x^2 + 64)^2 using trigonometric substitution. This detailed guide simplifies the denominator and applies a substitution method to find the solution involving inverse trigonometric functions. Suitable for undergraduate calculus students.

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