Show that the trigonometric identity
sin2x1+tan2x≡2sinxcos3x \frac{\sin 2x}{1 + \tan^2 x} \equiv 2 \sin x \cos^3 x 1+tan2xsin2x≡2sinxcos3xis valid for all x x\,x where the expression is defined.
In a wave mechanics simulation, the rate of change of energy density E E\,E with respect to phase ϕ\phiϕ (in radians) is modeled by the equation:
dEdϕ=30sin6ϕ1+tan23ϕ \frac{dE}{d\phi} = \frac{30 \sin 6\phi}{1 + \tan^2 3\phi} dϕdE=1+tan23ϕ30sin6ϕHence, determine the general expression for E(ϕ)E(\phi)E(ϕ).
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions 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), matched to the AQA A Level Maths (7357) 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.
