How to calculate $\frac{1}{2} \int_0^1\ 1.5 e^{-ik\pi \ t} \ \ dt, \, k \in \mathbb{Z} $

Question

$$\frac{1}{2} \int_0^1\ 1.5 e^{-ik\pi\ t} \ \ dt, \, k \in \mathbb{Z} $$

Answer 1

The computation is straightforward: \begin{align} \int_0^1 \frac34 e^{-ik\pi t}\ \mathsf {dt} &= \left.\frac{3i}{4k\pi}e^{-ik\pi t}\right|_0^1\\ &=\frac{3i}{4k\pi}(e^{-ik\pi}-1)\\ &=\frac3{4k\pi}((-1)^k-1)i. \end{align}