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

From part (b) we have the formula $c_{n} = \int_{0}^{1} f (t) e^{- 2 π i n t} d t .$ Plugging in n = 0 we calculate

\begin{aligned} c_{0} & = \int_{0}^{1} f (t) e^{- 2 π i (0) t} d t \\ = \int_{0}^{1} (t^{2} - t) (1) d t \\ = \frac{1}{3} - \frac{1}{2} = - \frac{1}{6} \end{aligned}