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.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.