f(x) = do + -San COS + bn sin 2an (2) n=1 n=1 do = 2 L f(x) dx , (3) an = J." f(x) cos ("TT .) da, n= 1,2,. (4) on = ["f(x) sin (" Trz ) de, n= 1,2,.. (5) (2) f(x) is neither even nor odd in this problem, so all coefficients may be non-zero. (3) These results of integration by parts may be helpful: 2 cos ar de = 2ac cos ax + sin ar + C, (6) a x2 sin andx = P N 72 sin ar cos ar + C. (7) a (8) The arbitrary constant C is irrelevant in this problem because the integrals are def- inite. Optional: You may type some of you Fourier terms in this website (4) Choose for instance L = n and x = /2 to get an expression of 12. Notice that at the extremes, x = 0, L the function is discontinuous. Hence, it is better to avoid such points to estimate the value of f(x) with the Fourier series. A value in the middle will also have a better convergence. Optional: Once you are done, you may use a calculator to introduce the first terms of your series of rational numbers in order to double check that your result approaches . You cannot use the calculator to derive the answer.2. 72 and rational numbers (40 pts) (a) Find the Fourier series of the function f(x) = x2,0