Q2Advanced Engineering Mathematics

The problem demands the calculation of both the Fourier Sine and Fourier Cosine transforms for a specific piecewise-defined spatial function $f(x)$. The function is defined as $f(x)=x$ for $0\le x\le 1$, and $f(x)=2-x$ for $1\le x\le 2$. The function is identically zero outside this bounded interval. Because the function is defined differently across two distinct intervals, any integration over its domain must be meticulously split into two separate definite integrals.

1. Evaluating the Fourier Sine Transform

The mathematical definition of the Fourier Sine Transform is $F_s(s)={\int }_0^{\infty }f(x)\sin (sx)dx$. We split this infinite integral according to the defined piecewise intervals:

Both integrals strictly require the advanced technique of integration by parts, governed by the formula $\int udv=uv-\int vdu$. For the first integral, we select $u=x,dv=\sin (sx)dx$, leading to $v=-\frac{\cos (sx)}{s}$. The evaluation yields ${\left[-\frac{x\cos (sx)}{s}+\frac{\sin (sx)}{s^2}\right]}_0^1=-\frac{\cos (s)}{s}+\frac{\sin (s)}{s^2}$.

For the second distinct integral, we select $u=(2-x),dv=\sin (sx)dx$. After rigorous evaluation from bounds 1 to 2, and adding the result to the first integral's outcome, many trigonometric terms will strategically cancel out, yielding a finalized, compact frequency-domain representation.

2. Evaluating the Fourier Cosine Transform

Similarly, the definition of the Fourier Cosine Transform is $F_c(s)={\int }_0^{\infty }f(x)\cos (sx)dx$. Again, we must partition the integration domain:

We employ integration by parts once more. For the first term, $u=x,dv=\cos (sx)dx$, resulting in $v=\frac{\sin (sx)}{s}$. The evaluation yields ${\left[\frac{x\sin (sx)}{s}+\frac{\cos (sx)}{s^2}\right]}_0^1=\frac{\sin (s)}{s}+\frac{\cos (s)}{s^2}-\frac{1}{s^2}$.

The second term is evaluated identically using $u=(2-x)$. Combining and mathematically simplifying the bounds from both distinct integrals provides the final exact expression for the Fourier Cosine Transform $F_c(s)$. This piecewise approach ensures no spatial data is lost during the transformation to the frequency domain.