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To solve the given integral:

\[
\int_0^{1/2} \sin(2\pi x) \, dx
\]

we can proceed as follows:

### Step 1: Set up the Integral
The integral is:

\[
\int_0^{1/2} \sin(2\pi x) \, dx
\]

### Step 2: Integrate \( \sin(2\pi x) \)
The antiderivative of \( \sin(2\pi x) \) with respect to \( x \) is:

\[
-\frac{1}{2\pi} \cos(2\pi x)
\]

So, we have:

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

### Step 3: Apply the Limits
Now we evaluate this from \( x = 0 \) to \( x = \frac{1}{2} \):

\[
\int_0^{1/2} \sin(2\pi x) \, dx = \left[ -\frac{1}{2\pi} \cos(2\pi x) \right]_0^{1/2}
\]

Substitute \( x = \frac{1}{2} \) and \( x = 0 \):

1. For \( x = \frac{1}{2} \):
   \[
   -\frac{1}{2\pi} \cos\left(2\pi \cdot \frac{1}{2}\right) = -\frac{1}{2\pi} \cos(\pi) = -\frac{1}{2\pi} \cdot (-1) = \frac{1}{2\pi}
   \]

2. For \( x = 0 \):
   \[
   -\frac{1}{2\pi} \cos(2\pi \cdot 0) = -\frac{1}{2\pi} \cos(0) = -\frac{1}{2\pi} \cdot 1 = -\frac{1}{2\pi}
   \]

### Step 4: Calculate the Result
Now, we subtract the values:

\[
\frac{1}{2\pi} - \left(-\frac{1}{2\pi}\right) = \frac{1}{2\pi} + \frac{1}{2\pi} = \frac{1}{\pi}
\]

### Final Answer:
The correct answer is:

\[
\boxed{\frac{1}{\pi}}
\]

So, the correct option is \( \text{d. } \frac{1}{\pi} \).

Would you like more details on this solution, or do you have any questions?

### Related Questions:
1. How do we find the antiderivative of trigonometric functions like \( \sin \) and \( \cos \)?
2. Why does \( \cos(\pi) \) equal \(-1\)?
3. How does the Fundamental Theorem of Calculus apply to definite integrals?
4. What are the basic properties of definite integrals over symmetric intervals?
5. How would the result change if the limits of integration were different?

**Tip:** Remember that trigonometric integrals often involve using the antiderivatives of \( \sin \) and \( \cos \), which are \( -\cos(x) \) and \( \sin(x) \), respectively.

Calculate the following definite integral \( \int_0^{1/2} \sin(2\pi x) \, dx \).

Learn how to calculate the definite integral \( \int_0^{1/2} \sin(2\pi x) \, dx \) with a detailed step-by-step solution. This problem involves trigonometric integration and applying the Fundamental Theorem of Calculus. Suitable for students in Grades 11-12.

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