The integral provided appears to be:

$\int \left( 4x^3 - 2\sqrt{x} + 8x + 10 \right) \, dx$

Let’s evaluate this integral term by term:

Step 1: Integrate $4x^3$

$\int 4x^3 \, dx = 4 \cdot \frac{x^4}{4} = x^4$

Step 2: Integrate $-2\sqrt{x}$

Rewrite $\sqrt{x}$ as $x^{1/2}$: $\int -2x^{1/2} \, dx = -2 \cdot \frac{x^{3/2}}{3/2} = -\frac{4}{3}x^{3/2}$

Step 3: Integrate $8x$

$\int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2$

Step 4: Integrate $10$

$\int 10 \, dx = 10x$

Combine Results

Putting it all together, we get: $\int \left( 4x^3 - 2\sqrt{x} + 8x + 10 \right) \, dx = x^4 - \frac{4}{3}x^{3/2} + 4x^2 + 10x + C$

where $C$ is the constant of integration.

Would you like more details on any step, or have any questions?

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Here are 5 additional questions related to this problem:

1. How would the result change if we had an integral of $4x^3 + 2\sqrt{x} + 8x - 10$ instead?
2. What techniques can be used to integrate functions involving higher roots, such as $\sqrt[3]{x}$?
3. How can you verify the solution to an integral by differentiating the result?
4. What are some practical applications of integrating polynomial functions like this one?
5. How would we approach this problem if limits were specified for a definite integral?

Tip: Always remember to rewrite roots as fractional exponents when integrating; it simplifies the process significantly.