In a signal processing model, the power intensity P P\,P at time t t\,t is given by P(t)=t2e−2tP(t) = t^2 e^{-2t}P(t)=t2e−2t. Determine the general integral for the energy flux, ∫t2e−2t dt\int t^2 e^{-2t} \, dt∫t2e−2tdt, providing the result in its simplest factorised form.
In a chemical reaction, the rate of change of a concentration with respect to a spatial coordinate r r\,r is modeled by the function f(r)=4r+1(r+1)(2r+1)\displaystyle f(r) = \frac{4r+1}{(r+1)(2r+1)}f(r)=(r+1)(2r+1)4r+1. Use partial fractions and algebraic integration to prove that
∫134r+1(r+1)(2r+1) dr=lnk \int_{1}^{3} \frac{4r+1}{(r+1)(2r+1)} \, dr = \ln k ∫13(r+1)(2r+1)4r+1dr=lnkwhere k k\,k is a rational constant to be determined.
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.