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.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.