## Answers

f(x) = x(4 - x) on the interval [0, 4] we are now going to do is to try to find a series representation for f(x) of the form f(x) = Sigma_n = 1^infinity B_n sin (n pi x/4) we know that B_m = 2/4 integral_0^4 f(x) sin (m pi x/4) dx for m = 1, 2, 3, ... = 1/2 integral_0^4 (4x - x^2) sin (m pi x/4) dx = 2 integral_0^4 x sin (m pi x/4) - 1/2 integral_0^4 x^2 sin (m pi x/4) dx Integration by parts = 2 [4/m pi x {-cos m pi x/4}]_0^4 + 2 middot 4/m pi middot 4/m pi [sin m pi x/4]_0^4 - 1/2 [(4/m pi)^2 {-(m pi x/4)^2 cos (m pi x/4) + 2 middot m pi x/4 middot sin (m pi x/4) + 2 cos (m pi x/4)}]_0^4 = -32/m pi cos (m pi)

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