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1.11 H: Integration

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Question 48

A structural engineer models the rate of change of deflection y y\,y of a loaded beam using the gradient function:

dydx=7x+1(x−2)(2x+1)2,x>2 \frac{dy}{dx} = \frac{7x+1}{(x-2)(2x+1)^2}, \quad x > 2 dxdy​=(x−2)(2x+1)27x+1​,x>2
a.

Find the values of the constants AAA, B B\,B and C C\,C such that

7x+1(x−2)(2x+1)2≡Ax−2+B2x+1+C(2x+1)2 \frac{7x+1}{(x-2)(2x+1)^2} \equiv \frac{A}{x-2} + \frac{B}{2x+1} + \frac{C}{(2x+1)^2} (x−2)(2x+1)27x+1​≡x−2A​+2x+1B​+(2x+1)2C​
[4]
b.

Hence find the exact change in deflection between x=3x=3x=3 and x=4x=4x=4 by calculating

∫347x+1(x−2)(2x+1)2 dx \int_{3}^{4} \frac{7x+1}{(x-2)(2x+1)^2} \, \mathrm{d}x ∫34​(x−2)(2x+1)27x+1​dx

giving your answer in the form pln⁡q+rp \ln q + rplnq+r where ppp, q q\,q and r r\,r are rational numbers.

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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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