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1.11 H: Integration

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Question 63
a.

Show that the trigonometric identity

sin⁡2x1+tan⁡2x≡2sin⁡xcos⁡3x \frac{\sin 2x}{1 + \tan^2 x} \equiv 2 \sin x \cos^3 x 1+tan2xsin2x​≡2sinxcos3x

is valid for all x x\,x where the expression is defined.

[3]
b.

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=30sin⁡6ϕ1+tan⁡23ϕ \frac{dE}{d\phi} = \frac{30 \sin 6\phi}{1 + \tan^2 3\phi} dϕdE​=1+tan23ϕ30sin6ϕ​

Hence, determine the general expression for E(ϕ)E(\phi)E(ϕ).

[6]
Markscheme

1.11 H: Integration Questions

  1. A Level
  2. /Maths
  3. /1.11 H: Integration

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.

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