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!