https://file.aiquickdraw.com/mathbot/file/1728529175244zfkgwj81.jpg

The integral provided in the image is:

\[
\int \frac{9 \, dx}{x \left(x^4 + 4\right)}
\]

### Step-by-step solution:

We can solve this integral using substitution. First, let’s try to make a substitution that simplifies the expression \(x^4 + 4\). Set:

\[
u = x^4 + 4
\]

Then, differentiate \(u\) with respect to \(x\):

\[
du = 4x^3 \, dx
\]

We notice, however, that this substitution doesn't directly cancel the \(x\) terms in the denominator easily. Instead, we can apply partial fractions or a more appropriate trigonometric substitution. A substitution \(x^2 = t\) can help transform the powers of \(x\) more directly.

Let me complete this solution by working through the calculation.

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Would you like more details or have any questions so far? 

Here are five related questions you might find interesting:
1. How do we use substitution techniques to simplify integrals?
2. How can we approach integrals involving polynomial functions in the denominator?
3. What role does the constant of integration play in indefinite integrals?
4. How do you identify when to use partial fraction decomposition in integration?
5. Can you explain why absolute values may be necessary in integrals?

**Tip:** In integrals involving polynomials, trigonometric substitution or algebraic manipulation can often help simplify expressions.

Evaluate the integral: ∫ (9 dx) / [x(x^4 + 4)]

Solve the integral ∫ (9 dx) / [x(x^4 + 4)] using substitution techniques. Learn how to approach integrals involving polynomials and rational functions with detailed steps. Suitable for undergraduate calculus students.

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