To find the particular solution for the anti-derivative of the function $f(x) = 2x^3 + 1 - \sin(x)$, given that $F(0) = 1$, we'll follow these steps:

Step 1: Find the General Antiderivative

To obtain the antiderivative $F(x)$, we integrate $f(x)$ with respect to $x$:

$F(x) = \int \left(2x^3 + 1 - \sin(x)\right) \, dx$

We can break this down into the sum of three separate integrals:

$F(x) = \int 2x^3 \, dx + \int 1 \, dx - \int \sin(x) \, dx$

Step 2: Integrate Each Term

Now, let's integrate each term individually:

1. Integrating $2x^3$: $\int 2x^3 \, dx = \frac{2x^4}{4} = \frac{x^4}{2}$

2. Integrating $1$: $\int 1 \, dx = x$

3. Integrating $-\sin(x)$: $\int -\sin(x) \, dx = \cos(x)$

So, combining these results, the general form of the antiderivative is:

$F(x) = \frac{x^4}{2} + x + \cos(x) + C$

where $C$ is the constant of integration.

Step 3: Determine the Constant $C$

To find the particular solution, we use the condition $F(0) = 1$. Substituting $x = 0$ into the general form:

$F(0) = \frac{(0)^4}{2} + (0) + \cos(0) + C = 1$

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

$1 + C = 1$

So, $C = 0$.

Final Answer

The particular solution for the antiderivative is:

$\boxed{F(x) = \frac{x^4}{2} + x + \cos(x)}$

This function satisfies the initial condition $F(0) = 1$.

---

Would you like any further details or have any questions? Here are some related questions you might find helpful:

1. How would the antiderivative change if $f(0)$ were a different value?
2. What is the derivative of $F(x)$ to verify it's indeed the antiderivative of $f(x)$?
3. Can you find $F(1)$ given the function $F(x)$?
4. How would you handle an initial condition where $f(x)$ has a trigonometric function like $\cos(x)$ instead of $\sin(x)$?
5. What does the constant $C$ represent in the context of indefinite integrals?

Tip: Always double-check the boundary or initial conditions when solving for the constant in an indefinite integral to ensure the particular solution satisfies the given condition.