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1.9 Calculus

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Question 128
a.

Determine the indefinite integral ∫ln⁡(x2)x3 dx\int \frac{\ln(x^2)}{x^3} \, \text{d}x∫x3ln(x2)​dx.

[3]
b.

A mechanical piston is subject to a variable resistive force F(x)F(x)F(x), where xxx is the displacement in metres from the start of the stroke. The force, in Newtons, is given by

F(x)=3+2x2+ln⁡(x2)x3,x≥1 F(x) = \frac{3 + 2x^2 + \ln(x^2)}{x^3}, \quad x \ge 1 F(x)=x33+2x2+ln(x2)​,x≥1

The work done by the force as the piston moves from x=1x = 1x=1 to x=2x = 2x=2 is given by ∫12F(x) dx\int_1^2 F(x) \, \text{d}x∫12​F(x)dx.

Using the result from part (a), find the exact work done, writing your answer in the form a+ln⁡ba + \ln ba+lnb, where aaa and bbb are constants to be determined.

[5]
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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