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The integral given is:

\[
\int \frac{x^2 + 9}{x^4 + 81} \, dx
\]

You have already started simplifying the integral, breaking it down as:

\[
\int \frac{1 + \frac{9}{x^2}}{x^2 + \frac{81}{x^2}} \, dx
\]

Now, let's proceed to solve it step by step.

### Substituting \( u = x^2 \)

Let’s perform the substitution \( u = x^2 \), hence \( du = 2x \, dx \) or \( dx = \frac{du}{2x} \). Also, since \( u = x^2 \), the expression becomes:

\[
\int \frac{1 + \frac{9}{x^2}}{x^2 + \frac{81}{x^2}} \, dx
= \int \frac{1 + \frac{9}{u}}{u + \frac{81}{u}} \, dx
\]

Simplify the denominator and numerator:

\[
= \int \frac{1 + \frac{9}{u}}{u + \frac{81}{u}} \, du
= \int \frac{u + 9}{u^2 + 81} \, du
\]

### Breaking the integral into two parts

Now, split the integral into two simpler integrals:

\[
\int \frac{u}{u^2 + 81} \, du + \int \frac{9}{u^2 + 81} \, du
\]

#### First integral: \( \int \frac{u}{u^2 + 81} \, du \)

For the first part, use the substitution \( v = u^2 + 81 \), so \( dv = 2u \, du \). This simplifies to:

\[
\frac{1}{2} \ln(u^2 + 81)
\]

#### Second integral: \( \int \frac{9}{u^2 + 81} \, du \)

For the second part, recognize this as a standard arctangent form:

\[
\int \frac{1}{u^2 + a^2} \, du = \frac{1}{a} \arctan\left( \frac{u}{a} \right)
\]

Thus:

\[
\int \frac{9}{u^2 + 81} \, du = \frac{9}{81} \arctan\left( \frac{u}{9} \right) = \frac{1}{9} \arctan\left( \frac{u}{9} \right)
\]

### Final result

Combining the results from the two integrals:

\[
\frac{1}{2} \ln(u^2 + 81) + \frac{1}{9} \arctan\left( \frac{u}{9} \right) + C
\]

Substitute back \( u = x^2 \) to get the final answer:

\[
\frac{1}{2} \ln(x^4 + 81) + \frac{1}{9} \arctan\left( \frac{x^2}{9} \right) + C
\]

This is the solution to the integral. 

Do you have any questions or need further clarification?

Here are 5 related questions:
1. What are some common techniques for solving integrals involving rational functions?
2. How do you identify when to use the substitution method in integration?
3. What is the geometric interpretation of the arctangent function in calculus?
4. How can partial fraction decomposition help with integrating rational expressions?
5. What are some alternative methods to solve integrals involving quadratics in the denominator?

**Tip:** When dealing with integrals, always check if a substitution can simplify the expression before attempting more complex techniques.

Learn how to solve the integral of (x^2 + 9) / (x^4 + 81) using substitution and standard integral formulas. The solution involves breaking the expression into simpler integrals and applying the arctangent and logarithmic integration techniques. Suitable for university-level students studying calculus.

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