Chapter 14.5, Problem 58E

(a)

To determine

To calculate: The value of $F\left(x\right)$ for $f\left(t\right)=1$ and for $f\left(t\right)=t$ if $F\left(x\right)={\displaystyle {\int }_0^{\infty }f\left(t\right)e^{-xt}dt}$.

(b)

To determine

To calculate: The formula for $F\left(x\right)$ if $f\left(t\right)=t^n\left(n=1,2,\mathrm{3...}\right)$ and $F\left(x\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(t\right)e^{-xt}dt}$.

(c)

To determine

To calculate: The formula for $F\left(x\right)$ if $f\left(t\right)=e^{at}$ and $F\left(x\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(t\right)e^{-xt}dt}$.