Given that
5x+3(x−1)(3x+1)2≡Ax−1+B3x+1+C(3x+1)2 \frac{5x+3}{(x-1)(3x+1)^2} \equiv \frac{A}{x-1} + \frac{B}{3x+1} + \frac{C}{(3x+1)^2} (x−1)(3x+1)25x+3≡x−1A+3x+1B+(3x+1)2Cfind the values of the constants AAA, BBB and CCC.
Hence find the exact value of
∫235x+3(x−1)(3x+1)2 dx \int_{2}^{3} \frac{5x+3}{(x-1)(3x+1)^2} \,\mathrm{d}x ∫23(x−1)(3x+1)25x+3dxgiving your answer in the form plnq+rp \ln q + rplnq+r where ppp, qqq and rrr 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.