Science:Math Exam Resources/Courses/MATH307/December 2012/Question 04 (c)/Solution 1

From part (b) we have the formula $c_{n}=\int _{0}^{1}f(t)e^{-2\pi int}\,dt.$ Plugging in n = 0 we calculate

${\begin{aligned}c_{0}&=\int _{0}^{1}f(t)e^{-2\pi i(0)t}\,dt\\&=\int _{0}^{1}(t^{2}-t)(1)dt\\&={\frac {1}{3}}-{\frac {1}{2}}=-{\frac {1}{6}}\end{aligned}}$