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The image presents a question asking to find the value of \( a_1 \) for the function \( f(x) = x \sin x \) within the interval \( (0, 2\pi) \).

Given that the question refers to \( a_1 \), this is likely related to the Fourier series expansion of the function. In a Fourier series, the coefficients \( a_1 \) (for the cosine terms) can be calculated using the formula for Fourier coefficients:

\[
a_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \cos(x) \, dx
\]

For the given function \( f(x) = x \sin x \), we would substitute this into the formula and evaluate the integral to find \( a_1 \).

Let's set up and compute the following integral for \( a_1 \):

\[
a_1 = \frac{1}{\pi} \int_0^{2\pi} x \sin(x) \cos(x) \, dx
\]

We can simplify the product \( \sin(x) \cos(x) \) using a trigonometric identity:

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

Thus, the integral becomes:

\[
a_1 = \frac{1}{2\pi} \int_0^{2\pi} x \sin(2x) \, dx
\]

Now, I will compute this integral to find the value of \( a_1 \).The value of the Fourier coefficient \( a_1 \) is \( -\frac{1}{2} \).

Thus, the correct answer is option **(b)** \( a_1 = -\frac{1}{2} \).

Would you like a more detailed explanation of how this result was obtained or have any further questions?

Here are five related questions for further exploration:
1. How do you calculate the Fourier coefficients for a given periodic function?
2. What is the physical significance of Fourier series in signal processing?
3. Can the Fourier series of \( x \sin x \) converge uniformly in the given interval?
4. How does the trigonometric identity for \( \sin(x)\cos(x) \) simplify other integrals?
5. What are the differences between sine and cosine Fourier series?

**Tip:** When computing integrals involving products of sine and cosine, trigonometric identities can often simplify the expression significantly.

If f(x) = x sin(x) in (0, 2π), then what is the value of a₁? Options are: (a) a₁ = 1/2, (b) a₁ = -1/2, (c) a₁ = 1/4, (d) a₁ = -1/4.

This problem involves calculating the Fourier coefficient a₁ for the function f(x) = x sin(x) within the interval (0, 2π). By applying the Fourier series formula and simplifying using trigonometric identities, the correct value for a₁ is determined as -1/2.

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