The Laplace transform F(x) of a function f(t) is given by the formula. $F ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} t$
Find F(x) if $f ( t ) = e ^ { n t }$ (n constant).

A) $\frac { 1 } { x }$
B) $\frac { 1 } { x - n }$
C) $\frac { n } { x }$
D) $\frac { n } { x - \mathrm { n } }$
E) $n x - n$

Question

The Laplace transform F(x) of a function f(t) is given by the formula.   $F ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} t$  
Find F(x) if $f ( t ) = e ^ { n t }$ (n constant).
 
A) $\frac { 1 } { x }$
B) $\frac { 1 } { x - n }$
C) $\frac { n } { x }$
D) $\frac { n } { x - \mathrm { n } }$
E) $n x - n$

Quizplus · Accepted Answer

The Laplace transform of $f(t) = e^{nt}$ is given by $F(x) = \int_{0}^{+\infty} e^{nt}e^{-xt} dt$. Simplifying the integrand gives $e^{(n-x)t}$, and integrating this expression with respect to $t$ from 0 to $\infty$ results in $\frac{1}{x-n}$, assuming $x > n$ for convergence.

The Laplace Transform F(x) of a Function F(t) Is Given F(x)=∫0+∞f(t)e−xt dtF ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} tF(x)=∫0+∞​f(t)e−xt dt

The Laplace Transform F(x) of a Function F(t) Is Given $F ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} t$