When to use half period and when use full period for fourier series coefficients

Question

In wikipedia the formula of An and Bn coefficients are integrals on the full period of the function. but here the coefficient of fourier series of sin(x) is calculated by integrating on half of period. I'm totally confused what should i use as boundary of coefficient integral?

Answer 1

If you have an even or even function, twice the half-period integral equals the full-period integral. Use the half-period integral whenever the function is defined only on $0$ to $L$, so you have the freedom to extend it to an odd or even function (depending on the application.)