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>2Find 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)2CHence 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+1dxgiving your answer in the form plnq+rp \ln q + rplnq+r where ppp, q q\,q and r r\,r are rational numbers.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) 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.
