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.
480 exam-style questions on AQA A Level Maths 1.11 H: Integration, covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.