We are given the second derivative of the function $f(x)$ as:

$f''(x) = 5x + 8\sin(x)$

and the initial conditions:

$f(0) = 4, \quad f'(0) = 2.$

To find $f(x)$, we will integrate the second derivative step by step.

Step 1: Find the first derivative $f'(x)$

Since $f''(x) = 5x + 8\sin(x)$, we integrate it to get $f'(x)$:

$f'(x) = \int (5x + 8\sin(x)) \, dx.$

We perform the integration:

$\int 5x \, dx = \frac{5x^2}{2}, \quad \int 8\sin(x) \, dx = -8\cos(x).$

Thus, we have:

$f'(x) = \frac{5x^2}{2} - 8\cos(x) + C_1,$

where $C_1$ is a constant of integration.

Step 2: Apply the initial condition $f'(0) = 2$

To determine $C_1$, use the initial condition $f'(0) = 2$:

$f'(0) = \frac{5(0)^2}{2} - 8\cos(0) + C_1 = 2.$

Since $\cos(0) = 1$, this simplifies to:

$-8 + C_1 = 2 \quad \Rightarrow \quad C_1 = 10.$

Thus, the first derivative is:

$f'(x) = \frac{5x^2}{2} - 8\cos(x) + 10.$

Step 3: Find the function $f(x)$

Next, we integrate $f'(x)$ to get $f(x)$:

$f(x) = \int \left( \frac{5x^2}{2} - 8\cos(x) + 10 \right) \, dx.$

Performing the integration:

$\int \frac{5x^2}{2} \, dx = \frac{5x^3}{6}, \quad \int -8\cos(x) \, dx = -8\sin(x), \quad \int 10 \, dx = 10x.$

Thus, we have:

$f(x) = \frac{5x^3}{6} - 8\sin(x) + 10x + C_2,$

where $C_2$ is a constant of integration.

Step 4: Apply the initial condition $f(0) = 4$

To determine $C_2$, use the initial condition $f(0) = 4$:

$f(0) = \frac{5(0)^3}{6} - 8\sin(0) + 10(0) + C_2 = 4.$

Since $\sin(0) = 0$, this simplifies to:

$C_2 = 4.$

Final Solution

Thus, the function $f(x)$ is:

$f(x) = \frac{5x^3}{6} - 8\sin(x) + 10x + 4.$

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Would you like more details on any step, or do you have any further questions? Here's a list of related questions:

1. How would we compute the third derivative $f'''(x)$ from the given function?
2. What happens to $f(x)$ if the initial conditions change?
3. How can we determine the behavior of $f(x)$ as $x \to \infty$?
4. How can we use Taylor series expansion to approximate $f(x)$ around $x = 0$?
5. What is the significance of the term $-8\sin(x)$ in the context of the second derivative?

Tip: When given a second-order differential equation, always integrate step by step and apply the initial conditions early to simplify solving.