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).
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions 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, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
