Find the particular solution for the anti-derivative of f(x)=2x^3 +1-sin(x), if f(0)=1.

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.

Learn how to find the particular solution for the antiderivative of the function f(x)=2x^3 + 1 - sin(x) with a detailed step-by-step solution. This problem involves integrating polynomials and trigonometric functions, determining the constant of integration, and verifying the initial condition f(0)=1. Suitable for advanced high school and college-level students.

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