The expression you're referring to involves calculating the Fourier coefficients, specifically for the function $f(x) = \sin(nx)$ over the interval $[0, \pi]$. Here's how we proceed step-by-step:

Step-by-step breakdown:

1. Fourier Coefficient Formula: The formula for the Fourier sine coefficient $b_n$ is given by: $b_n = \frac{2}{\pi} \int_0^\pi f(x) \sin(nx) \, dx$ In this case, we are dealing with the function $f(x) = \sin(x)$.

2. Set up the integral: We substitute $f(x) = \sin(x)$ into the formula for $b_n$: $b_n = \frac{2}{\pi} \int_0^\pi \sin(x) \sin(nx) \, dx$

3. Use the Product-to-Sum Formula: To simplify the product of sines, we use the identity: $\sin(x) \sin(nx) = \frac{1}{2} \left[ \cos((n-1)x) - \cos((n+1)x) \right]$ This allows us to split the integral into two parts: $b_n = \frac{2}{\pi} \int_0^\pi \frac{1}{2} \left[ \cos((n-1)x) - \cos((n+1)x) \right] dx$

4. Simplify the Integral: Now, calculate the integral term-by-term: $b_n = \frac{1}{\pi} \left[ \int_0^\pi \cos((n-1)x) \, dx - \int_0^\pi \cos((n+1)x) \, dx \right]$

5. Evaluate the Integrals: Each of these integrals is straightforward: $\int_0^\pi \cos(kx) \, dx = \frac{\sin(kx)}{k} \Big|_0^\pi = \frac{\sin(k\pi)}{k} - \frac{\sin(0)}{k}$ Since $\sin(k\pi) = 0$ for any integer $k$, both integrals evaluate to 0 unless $k = 0$.

6. Conclusion: The integrals evaluate to 0 unless $n = 1$ (since $\cos(nx)$ at $x = 0$ and $x = \pi$ gives us nonzero values only for odd multiples of $n$). Therefore, $b_n$ will only be non-zero for odd $n$, and it will have the value: $b_n = \frac{2}{n\pi} \quad \text{for odd } n$ For even $n$, $b_n = 0$.

Final result:

$b_n = \frac{2}{n\pi} \quad \text{for odd } n, \quad b_n = 0 \quad \text{for even } n$

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Would you like further clarification or more examples on Fourier series? Here are some related questions you could explore:

1. How do you calculate the Fourier series for other periodic functions like square waves or sawtooth waves?
2. Can you derive the Fourier sine and cosine series for a general function over an arbitrary interval?
3. How does the choice of orthogonal functions (like sine and cosine) help in Fourier analysis?
4. What is the significance of the orthogonality property in Fourier series expansions?
5. How does the convergence of the Fourier series work when the function is discontinuous?

Tip: When solving Fourier integrals, always check if you can use standard identities (like the product-to-sum identity) to simplify the integrals!