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 AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions 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), matched to the AQA A Level Maths (7357) 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.
