I am trying to learn PDEs by my self these days, because I need it for my personal projects; Thanks everyone helped me with my self-learning since yesterday.
I'm not sure my answer is correct for the following question; I would appreciate if you check it for me :)
I want to find Fourier cosine integral of:
\begin{cases} \frac{\pi}{2}\sin(x) & 0\leq x \leq \pi \\ 0 & x>\pi \end{cases}
$$ f(x)=\int_0^\infty A(\alpha)\cos(\alpha x)d\alpha = \int_0^\pi A(\alpha)\cos(\alpha x)d\alpha+\int_\pi^\infty A'(\alpha)\cos(\alpha x)d\alpha $$
$$ A(\alpha)=\frac{2}{\pi}\int_0^\pi\frac{\pi}{2}\sin(x)\cos(\alpha x)dx $$
$$ A'(\alpha)=0 $$
Is it correct?