The image shows the definite integral:

$\int_0^1 (4x^3 - 2x + 10) \, dx$

Let's solve this step by step:

Step 1: Integrate the function

We need to find the indefinite integral of $4x^3 - 2x + 10$.

$\int (4x^3 - 2x + 10) \, dx = \frac{4x^4}{4} - \frac{2x^2}{2} + 10x + C$

Simplifying the terms:

$x^4 - x^2 + 10x + C$

Step 2: Apply the definite integral bounds

Now, we need to evaluate the integral from $0$ to $1$:

$\left[ x^4 - x^2 + 10x \right]_0^1$

Evaluate at $x = 1$:

$1^4 - 1^2 + 10(1) = 1 - 1 + 10 = 10$

Evaluate at $x = 0$:

$0^4 - 0^2 + 10(0) = 0$

Step 3: Subtract the values

Now subtract the value at $x = 0$ from the value at $x = 1$:

$10 - 0 = 10$

Thus, the value of the definite integral is:

$\boxed{10}$

Would you like more details, or do you have any other questions?

Related Questions:

1. How do we handle definite integrals with more complex functions?
2. What is the difference between definite and indefinite integrals?
3. Can the bounds of a definite integral ever be negative?
4. How would the result change if the function involved trigonometric terms?
5. What applications does this kind of integral have in physics or engineering?

Tip:

Always simplify the function before applying the limits when working with definite integrals; this helps avoid errors.