Q3Advanced Engineering Mathematics I

Fourier Sine Transform: $F_s(s)=\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }f(x)\sin (sx)dx$

$=\sqrt{\frac{2}{\pi }}\left[{\int }_0^{1}x\sin (sx)dx+{\int }_1^2(2-x)\sin (sx)dx\right]$

Using integration by parts: $\int x\sin (sx)dx=-\frac{x}{s}\cos (sx)+\frac{1}{s^2}\sin (sx)$.

Fourier Cosine Transform: $F_c(s)=\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }f(x)\cos (sx)dx$

$=\sqrt{\frac{2}{\pi }}\left[{\int }_0^{1}x\cos (sx)dx+{\int }_1^2(2-x)\cos (sx)dx\right]$.