\begin{align*}
(\phi f)(p) &= \int_{0}^{\infty} \sin x \cos x ~ e^{-px} dx \\
&= \frac{1}{4i} \int_{0}^{\infty} (e^{2 i x} - e^{-2ix}) e^{-px} dx \\
&= \frac{1}{4i} \left( \int_{0}^{\infty} e^{(2i-p)x)} dx - \int_{0}^{\infty} e^{-(2i+p)x} dx \right) \\
&= \frac{1}{4i} \left( \frac{1}{2i-p} e^{(2i-p)x} \bigg|_{0}^{\infty} - \frac{1}{2i+p} e^{-(2i+p)x} \bigg|_{0}^{\infty} \right) \\
&= \frac{1}{4i} \left(\frac{1}{2i-p}(0-1) + \frac{1}{2i+p}(0-1) \right) \\
&= -\frac{1}{4i} \left(\frac{2i + p + 2i - p}{- 4 - p^2} \right) \\

maximal and prime ideals are very related concepts. the way to understand them, as far as I'm concerned, is by their quotients. prime ideals are precisely those ideals whose quotient is an integral domain and maximal ideals are those whose quotient is a field. this is how these concepts are often used in practice and it immediately makes clear why maximal ideals are prime.
principal ideals is a very different notion from those other two. they are simply ideals that can be generated by a single element. I don't think there's a more convenient way of thinking about them, they're simply very h…