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Fourier series expansion of periodic function

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Consider a periodic function f(x) with period L.
Over one period, f(x) = sin(2*pi*x/L) over the interval -L/4 to L/4, f(x) = 0 over the intervals -L/2 to -L/4, and L/4 to L/2.
Derive an expression for the nth Fourier series coefficient, an.
In the Fourier series expansion

plot for n = -4 to 4 using matlab, and superimpose onto a plot for f(x)

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The solution is comprised of detailed calculation of Fourier series expansion of periodic function. It also includes the Matlab codes to plot the graphs.

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