Evaluate the definite integral.
 
4𝜋       -4pi

(sin2 *x* + 1) *dx*

To evaluate the definite integral 

\[
\int_{-4\pi}^{4\pi} \left( \sin^2(x) + 1 \right) \, dx
\]

we can follow these steps:

### Step 1: Split the integral

The integrand is \( \sin^2(x) + 1 \). So we can split the integral as:

\[
\int_{-4\pi}^{4\pi} (\sin^2(x) + 1) \, dx = \int_{-4\pi}^{4\pi} \sin^2(x) \, dx + \int_{-4\pi}^{4\pi} 1 \, dx
\]

Now we evaluate each term separately.

### Step 2: Evaluate the second integral

\[
\int_{-4\pi}^{4\pi} 1 \, dx = (4\pi - (-4\pi)) = 8\pi
\]

### Step 3: Evaluate the first integral using symmetry

\( \sin^2(x) \) is an even function, meaning \( \sin^2(-x) = \sin^2(x) \). Therefore, the integral over \( [-4\pi, 4\pi] \) can be written as twice the integral over \( [0, 4\pi] \):

\[
\int_{-4\pi}^{4\pi} \sin^2(x) \, dx = 2 \int_0^{4\pi} \sin^2(x) \, dx
\]

Now, to integrate \( \sin^2(x) \), we use the identity:

\[
\sin^2(x) = \frac{1 - \cos(2x)}{2}
\]

Thus, the integral becomes:

\[
2 \int_0^{4\pi} \sin^2(x) \, dx = 2 \int_0^{4\pi} \frac{1 - \cos(2x)}{2} \, dx = \int_0^{4\pi} (1 - \cos(2x)) \, dx
\]

### Step 4: Evaluate the integral

The integral of \( 1 \) over \( [0, 4\pi] \) is:

\[
\int_0^{4\pi} 1 \, dx = 4\pi
\]

The integral of \( \cos(2x) \) is:

\[
\int_0^{4\pi} \cos(2x) \, dx = \frac{\sin(2x)}{2} \Big|_0^{4\pi} = \frac{\sin(8\pi) - \sin(0)}{2} = 0
\]

Thus, the integral simplifies to:

\[
\int_0^{4\pi} (1 - \cos(2x)) \, dx = 4\pi
\]

### Step 5: Combine results

Now we can combine everything:

\[
\int_{-4\pi}^{4\pi} \sin^2(x) \, dx = 2 \times 4\pi = 8\pi
\]

So the total integral is:

\[
\int_{-4\pi}^{4\pi} (\sin^2(x) + 1) \, dx = 8\pi + 8\pi = 16\pi
\]

### Final Answer:
The value of the definite integral is \( \boxed{16\pi} \).

---

Would you like more details on any step? 

Here are five related questions to consider:
1. How does the symmetry of functions simplify integrals over symmetric intervals?
2. Why is \( \sin^2(x) \) considered an even function?
3. What other trigonometric identities can help simplify integrals?
4. How would the result change if the limits were different, say from \( -\pi \) to \( \pi \)?
5. Can you apply similar techniques to other trigonometric integrals?

**Tip:** When integrating trigonometric functions, use symmetry and identities like \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) to simplify your work!

Evaluate the definite integral: ∫ from -4π to 4π of (sin²(x) + 1) dx

Learn how to evaluate the definite integral of (sin²(x) + 1) from -4π to 4π using integral calculus techniques. This problem involves splitting the integral, applying trigonometric identities, and using the symmetry of even functions. Detailed solution provided for college-level calculus students.

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