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).
864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.