Chapter A, Problem 55E

(a)

To determine

The value of $F^{\prime }\left(x\right)$, if $F\left(x\right)={\displaystyle {\int }_0^{\infty }\frac{1-e^{-x{t}^2}}{t^2}}dt$ 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 $F^{\prime }\left(x\right)=\frac{\sqrt{\pi }}{2\sqrt{x}}$.

(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 $F\left(0\right)=0$.