To find the most general antiderivative (indefinite integral) of the given function:

$3x^4 - \frac{5}{x^5} - 3,$

we integrate each term with respect to $x$. Here's the solution step by step:

Step 1: Rewrite the function

For easier integration, rewrite the terms with $x$-exponents:

$3x^4 - \frac{5}{x^5} - 3 = 3x^4 - 5x^{-5} - 3.$

Step 2: Integrate each term

Use the power rule for integration:

\int x^n \, dx = \frac{x^{n+1}}{n+1} \quad \text{(for n \neq -1)}.

1. First term: $3x^4$:
   $\int 3x^4 \, dx = 3 \cdot \frac{x^{4+1}}{4+1} = 3 \cdot \frac{x^5}{5} = \frac{3}{5}x^5.$

2. Second term: $-5x^{-5}$:
   $\int -5x^{-5} \, dx = -5 \cdot \frac{x^{-5+1}}{-5+1} = -5 \cdot \frac{x^{-4}}{-4} = \frac{5}{4}x^{-4}.$

3. Third term: $-3$:
   $\int -3 \, dx = -3x.$

Step 3: Combine the results

Adding all the integrated terms, we get:

$\int \left( 3x^4 - 5x^{-5} - 3 \right) dx = \frac{3}{5}x^5 + \frac{5}{4}x^{-4} - 3x + C.$

Final Answer:

The antiderivative is:

$\boxed{\frac{3}{5}x^5 + \frac{5}{4}x^{-4} - 3x} + C.$

Let me know if you need any clarification!

Related Questions:

1. How do you handle integration when $n = -1$ in the power rule?
2. What is the general formula for finding an antiderivative?
3. How do you integrate functions involving fractions, such as $\frac{1}{x^2}$?
4. How can you check if your antiderivative is correct?
5. What is the role of the constant $C$ in indefinite integrals?

Tip:

To verify your solution, differentiate the antiderivative and ensure you obtain the original function.