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.
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.
