https://file.aiquickdraw.com/mathbot/file/1725563078784sltuvhio.jpg

The expression you provided in the image is:

\[
\int (3x^2 - \cos^2(x)) \, dx
\]

To solve this indefinite integral, we will approach it term by term.

### Step-by-Step Solution:

1. **Integrating \( 3x^2 \):**
   \[
   \int 3x^2 \, dx = x^3
   \]

2. **Integrating \( -\cos^2(x) \):**
   This term requires a trigonometric identity to simplify it. Use the identity:
   \[
   \cos^2(x) = \frac{1 + \cos(2x)}{2}
   \]
   So,
   \[
   -\int \cos^2(x) \, dx = -\int \frac{1 + \cos(2x)}{2} \, dx
   \]
   Split the integral:
   \[
   -\frac{1}{2} \int (1 + \cos(2x)) \, dx = -\frac{1}{2} \left( \int 1 \, dx + \int \cos(2x) \, dx \right)
   \]
   The integrals are straightforward:
   \[
   \int 1 \, dx = x
   \]
   \[
   \int \cos(2x) \, dx = \frac{\sin(2x)}{2}
   \]
   So,
   \[
   -\frac{1}{2} \left( x + \frac{\sin(2x)}{2} \right) = -\frac{x}{2} - \frac{\sin(2x)}{4}
   \]

### Final Answer:
Combining both parts:

\[
\int (3x^2 - \cos^2(x)) \, dx = x^3 - \frac{x}{2} - \frac{\sin(2x)}{4} + C
\]

Where \( C \) is the constant of integration.

---

Would you like more details or have any questions about the solution?  
Here are some related questions you might find interesting:

1. How do we handle trigonometric integrals that involve higher powers of cosine or sine?
2. Can you apply a similar method to integrals involving \(\sin^2(x)\)?
3. What are common trigonometric identities used in integrals?
4. How can substitution be used to solve more complex trigonometric integrals?
5. How would the result change if \(\cos^2(x)\) was replaced with \(\sin^2(x)\)?

**Tip:** When dealing with powers of trigonometric functions, it's often useful to use identities to simplify them before integrating.

Learn how to solve the indefinite integral ∫(3x^2 - cos^2(x)) dx using trigonometric identities and basic integration rules. A step-by-step guide, perfect for undergraduate students studying Calculus I.

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