The integral provided is:

$\int \left(4x^4 - 2x^3 + 7x - 4\right) \, dx$

Let's solve this step by step.

Step 1: Integrate each term individually

The integral of a polynomial can be done by integrating each term separately. The power rule for integration states that:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

Applying this to each term:

1. $\int 4x^4 \, dx = 4 \cdot \frac{x^{5}}{5} = \frac{4x^5}{5}$
2. $\int -2x^3 \, dx = -2 \cdot \frac{x^{4}}{4} = -\frac{2x^4}{4} = -\frac{x^4}{2}$
3. $\int 7x \, dx = 7 \cdot \frac{x^{2}}{2} = \frac{7x^2}{2}$
4. $\int -4 \, dx = -4x$

Step 2: Combine the results

Now, sum the results of the individual integrations:

$\int \left(4x^4 - 2x^3 + 7x - 4\right) \, dx = \frac{4x^5}{5} - \frac{x^4}{2} + \frac{7x^2}{2} - 4x + C$

Where $C$ is the constant of integration.

Final Answer:

$\frac{4x^5}{5} - \frac{x^4}{2} + \frac{7x^2}{2} - 4x + C$

Would you like any further details or explanations on this solution?

Here are 5 related questions that might interest you:

1. How do you handle definite integrals with polynomial functions?
2. What is the difference between indefinite and definite integrals?
3. How would the integral change if one of the terms had a fractional exponent?
4. Can you explain the fundamental theorem of calculus?
5. How do you solve integrals involving trigonometric functions?

Tip: Remember, when dealing with polynomial integrals, always apply the power rule carefully, and don't forget to add the constant of integration $C$ in indefinite integrals!