Determine the indefinite integral ∫e−tsin(2t) dt\int e^{-t} \sin(2t) \, dt∫e−tsin(2t)dt.
A surge of electrical power, P P\,P kilowatts, in an experimental circuit is modeled by the function
P(t)=e−tsin(2t),t≥0 P(t) = e^{-t} \sin(2t), \quad t \ge 0 P(t)=e−tsin(2t),t≥0where t t\,t is the time in seconds since the circuit was activated.
Show that the total energy EEE (in kilojoules) transferred during the first pulse of power, which occurs between t=0t = 0t=0 and the first time t>0 t > 0\,t>0 where P(t)=0P(t) = 0P(t)=0, is exactly 25(1+e−π/2)\displaystyle \frac{2}{5}(1 + e^{-\pi/2})52(1+e−π/2).