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The given expression involves evaluating the indefinite integral:

\[
\int \left( 5x^4 - \frac{4}{x^4} \right) dx
\]

Let's break this down and solve it step by step.

### Step 1: Integrating each term separately
We can integrate the terms \(5x^4\) and \(-\frac{4}{x^4}\) separately.

#### a) Integral of \( 5x^4 \):
\[
\int 5x^4 dx = 5 \cdot \frac{x^{5}}{5} = x^5
\]

#### b) Integral of \(-\frac{4}{x^4}\) (rewrite as \(-4x^{-4}\)):
\[
\int -4x^{-4} dx = -4 \cdot \frac{x^{-3}}{-3} = \frac{4}{3} x^{-3}
\]

### Step 2: Combining the results
Now, putting the integrals together:
\[
\int \left( 5x^4 - \frac{4}{x^4} \right) dx = x^5 + \frac{4}{3}x^{-3} + C
\]
where \(C\) is the constant of integration.

### Final Answer:
\[
x^5 + \frac{4}{3}x^{-3} + C
\]

Let me know if you would like further details or have any questions!

Here are 5 related questions to expand your understanding:
1. How do you apply the power rule to integrate functions of the form \(x^n\)?
2. What is the significance of the constant of integration in indefinite integrals?
3. How would you integrate functions involving trigonometric terms, such as \(\sin(x)\) or \(\cos(x)\)?
4. Can you explain how to integrate a rational function, such as \(\frac{1}{x^2+1}\)?
5. What are the differences between definite and indefinite integrals?

**Tip:** Always simplify your expressions before integrating, as it makes the process easier and reduces errors!

Evaluate the following indefinite integral: ∫(5x^4 - 4/x^4) dx

Learn how to solve the indefinite integral of the expression 5x^4 - 4/x^4. This calculus problem demonstrates the application of the power rule and simplifies negative exponents for integration. Suitable for students in Grades 11-12 and college-level calculus.

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