Answer to Question #345688 in Differential Equations for Wes

Question

Calculate the Laplace transform of

• (t) integral from (0 to t) (T)e^(T)dT

Answer 1

$${\cal L}\{f\}(s)=\int_0^{\infty}f(t)e^{-st}dt$$ $${\cal L}\{t\int_0^tTe^TdT\}(s)=\int_0^{\infty}t\int_0^tTe^TdTe^{-st}dt$$ $$=\int_0^{\infty}t(1+e^t(t-1))e^{-st}dt=\frac{3s-1}{s^2(s-1)^3}$$