Given that
2x2−x+1x+1≡Ax+B+Cx+1\displaystyle \frac{2x^2 - x + 1}{x + 1} \equiv Ax + B + \frac{C}{x + 1}x+12x2−x+1≡Ax+B+x+1C, x≠−1x \neq -1x=−1
find the values of the constants AAA, B B\,B and CCC.
Hence, using algebraic integration, find the exact value of
∫072x2−x+1x+1 dx\displaystyle\int_0^7 \frac{2x^2 - x + 1}{x + 1}\,dx∫07x+12x2−x+1dx
giving your answer in the form a+bln2a + b\ln 2a+bln2, where a a\,a and b b\,b are integers to be found.
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.