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.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.