Find
∫2x2e−4x dx \int 2x^2 e^{-4x} \, dx ∫2x2e−4xdxwriting the answer in simplest form.
The rate of change of the volume of water in a reservoir, VVV (in megalitres per hour), is modeled by the equation
dVdt=10t+2(t−1)(2t+1),t>1 \frac{dV}{dt} = \frac{10t+2}{(t-1)(2t+1)}, \quad t > 1 dtdV=(t−1)(2t+1)10t+2,t>1where t t\,t is the time in hours since a valve was opened. Use partial fractions and algebraic integration to show that the total change in volume between t=2t = 2t=2 and t=4t = 4t=4 is lnk \ln k\,lnk megalitres, where k k\,k is a fully simplified rational constant to be found.
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.