Chapter A, Problem 56E

(a)

To determine

The value of $F^{\prime }\left(x\right)$, if $F\left(x\right)={\displaystyle {\int }_0^{\infty }{e}^{-tx}\frac{\sin t}{t}}dt$ where $x\ge 0$.

(b)

To determine

The function $F\left(x\right)$ in the form $F\left(x\right)={\displaystyle \int F^{\prime }\left(x\right)}dx+C$ by the use of $F^{\prime }\left(x\right)=-\frac{1}{x^2+1}$.

(c)

To determine

The explicit function $F\left(x\right)$ in the form $F\left(x\right)={\displaystyle \int F^{\prime }\left(x\right)}dx+c$ by the result of sub-part (b) and $F\left(0\right)=\frac{\pi }{2}$.