Laplace Transform equation

Question

Equation

Viewing the picture attached, can anybody tell me why $$-\frac{A}{s+a}*e^{-(s+a)*t}$$ suddenly becomes $$\frac{A}{s+a}$$ in this example? What happened to our exponential equation and why is our fraction now positive instead of negative?

Answer 1

This is because $$ e^{-(s+a)t}\bigg|_{t=\infty}=\lim_{t\to\infty} e^{-(s+a)t}=0.$$ Update: Using $$ \int e^{ax}dx=\frac1a e^{ax}+C $$ one has \begin{eqnarray} F(s)&=&A\int_0^\infty e^{-(s+a)t}\\ &=&-\frac{A}{s+a}e^{-(s+a)t}\bigg|_{t=0}^\infty\\ &=&-\frac{A}{s+a}\bigg[\lim_{t\to\infty}e^{-(s+a)t}-1\bigg]\\ &=&\frac{A}{s+a} \end{eqnarray} if $s+a>0$.