Find the Fourier Transform of $F(x)\begin{cases} 1-x^2 & |x|<1 \\ 0 & |x|>1 \\ \end{cases} $
My workings so far: $f(\lambda) = \int_{-\infty}^{\infty} e^{-i\lambda x} f(x) \ dx = \int_{0}^{1} e^{-\lambda i x }(1-x^2) dx = (1-x^2)(\frac{e^{-xi \lambda}}{\lambda})+2x(\frac{e^{-xi \lambda}}{\lambda^2})-2(\frac{e^{-xi \lambda}}{\lambda^3})|_{0}^{1}$ which gives me the incorrect answer. The correct answer is $4(\frac{\lambda cos \lambda - sin \lambda}{\lambda^3})$
Please help!
Try adjusting the boundaries of your integral in the second equation. It should be \begin{equation} f(\lambda)=\int_{\mathbb{R}}e^{-i\lambda x}F(x)dx = \int_{-1}^1 e^{-i\lambda x}(1-x^2)dx, \end{equation} due to the Definition of F. The rest is just integration by parts.