2019-II-34

答案：D
$\begin{array}{cl}
& \dfrac{4 +i^5}{a + i} – i^6 \\
= &\dfrac{4 + i^4 \times i}{a + i} – i^4 \times i^2 \\
= & \dfrac{4 + i}{a + i} – (-1) \\
= & \dfrac{(4+i)(a-i)}{(a+i)(a-i)} + \dfrac{(a+i)(a-i)}{(a+i)(a-i)} \\
= & \dfrac{4a + ai – 4i -i^2 + a^2 – i^2}{a^2 – i^2} \\
= & \dfrac{(a^2 + 4a + 2) + (a- 4)i}{a^2 + 1} \\
= & \dfrac{a^2 + 4a + 2}{a^2 + 1} +\dfrac{a- 4}{a^2 + 1} i
\end{array}$

所以，該複數的實部為 $\dfrac{a^2 + 4a + 2}{a^2 + 1}$。