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1.9 Calculus

1.9 Calculus

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Question 370
i.

Determine ∫sin⁡2θ1+sin⁡θ dθ\int \frac{\sin 2\theta}{1 + \sin \theta} \, d\theta∫1+sinθsin2θ​dθ

[4]
ii.

A research team monitors the rate of energy consumption, EEE (in gigajoules per hour), of a large-scale data center. The rate is modeled by the function f(t)=18t2t+3f(t) = \frac{18t}{\sqrt{2t + 3}}f(t)=2t+3​18t​, where ttt is the time in hours since midnight.

Use the substitution u=2t+3u = \sqrt{2t + 3}u=2t+3​ to show that

∫18t2t+3 dt=2(2t+3)12(At+B)+k \int \frac{18t}{\sqrt{2t + 3}} \, dt = 2(2t + 3)^{\frac{1}{2}}(At + B) + k ∫2t+3​18t​dt=2(2t+3)21​(At+B)+k

where AAA and BBB are integers to be found and kkk is an arbitrary constant.

[6]
Markscheme

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

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.

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