To solve the given integral:

$\int \left( -3x^6 - \frac{3}{x} - \frac{4}{x^5} + 5\sqrt{x} \right) dx$

Step 1: Break the integral into individual terms

$\int \left( -3x^6 - \frac{3}{x} - \frac{4}{x^5} + 5\sqrt{x} \right) dx = \int -3x^6 \, dx - \int \frac{3}{x} \, dx - \int \frac{4}{x^5} \, dx + \int 5\sqrt{x} \, dx$

Step 2: Solve each term separately

1. For $-3x^6$: $\int -3x^6 \, dx = -\frac{3x^7}{7}$

2. For $-\frac{3}{x}$ (note that $\frac{1}{x} = x^{-1}$): $\int -\frac{3}{x} \, dx = -3 \ln|x|$

3. For $-\frac{4}{x^5}$ (rewrite as $-4x^{-5}$): $\int -\frac{4}{x^5} \, dx = -\frac{4x^{-4}}{-4} = x^{-4} = \frac{1}{x^4}$

4. For $5\sqrt{x}$ (rewrite as $5x^{1/2}$): $\int 5\sqrt{x} \, dx = \int 5x^{1/2} \, dx = \frac{5x^{3/2}}{3/2} = \frac{10x^{3/2}}{3}$

Step 3: Combine all results

Now add all the terms together: $\int \left( -3x^6 - \frac{3}{x} - \frac{4}{x^5} + 5\sqrt{x} \right) dx = -\frac{3x^7}{7} - 3 \ln|x| + \frac{1}{x^4} + \frac{10x^{3/2}}{3} + C$

Final Answer:

$\boxed{-\frac{3x^7}{7} - 3 \ln|x| + \frac{1}{x^4} + \frac{10x^{3/2}}{3} + C}$

Let me know if you want further details or clarification!

Related Questions:

1. What is the geometric interpretation of an indefinite integral?
2. How does the integration of fractional exponents like $x^{1/2}$ work?
3. What happens if the integral contains a trigonometric function instead of a polynomial?
4. How do you handle definite integrals for similar expressions?
5. What techniques are useful for integrating rational functions with complex denominators?

Tip: Always rewrite terms like $\sqrt{x}$ or $\frac{1}{x^n}$ into powers of $x$ for easier integration!