https://file.aiquickdraw.com/mathbot/file/17253946845966z7a5ltr.jpg

The image shows a definite integral:

\[
\int_{-\pi}^{\pi} \sin(x) \, dx
\]

To solve this integral, we'll evaluate it step by step.

### Step 1: Find the Antiderivative
The antiderivative of \(\sin(x)\) is \(-\cos(x)\).

\[
\int \sin(x) \, dx = -\cos(x)
\]

### Step 2: Apply the Limits of Integration
Now, we apply the limits \(-\pi\) and \(\pi\) to the antiderivative.

\[
\left[-\cos(x)\right]_{-\pi}^{\pi} = -\cos(\pi) - \left(-\cos(-\pi)\right)
\]

### Step 3: Evaluate the Cosine Function
We know that \(\cos(\pi) = -1\) and \(\cos(-\pi) = -1\). Substituting these values in:

\[
-\cos(\pi) - (-\cos(-\pi)) = -(-1) - (-(-1)) = 1 - 1 = 0
\]

### Final Answer
The value of the integral is:

\[
\int_{-\pi}^{\pi} \sin(x) \, dx = 0
\]

Would you like any more details or have any questions about this solution?

### Related Questions:
1. How would the integral change if the limits were from \(0\) to \(\pi\) instead of \(-\pi\) to \(\pi\)?
2. Can you explain why the integral of an odd function over a symmetric interval is always zero?
3. What would be the integral of \(\cos(x)\) over the same limits \([-π, π]\)?
4. How would this problem be solved if we were dealing with a different trigonometric function, like \(\sin^2(x)\)?
5. Could you show how to solve the integral of \(\sin(x)\) over a different interval, such as \([0, 2\pi]\)?

### Tip:
When dealing with integrals of trigonometric functions over symmetric intervals, it's useful to remember the properties of even and odd functions. This can often simplify the problem.

Evaluate the definite integral from -π to π of sin(x) dx.

Learn how to evaluate the definite integral of sin(x) from -π to π with a step-by-step solution. This problem covers key calculus concepts such as definite integrals, trigonometric functions, and the properties of odd functions over symmetric intervals. Suitable for students in Grades 11-12.

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