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.
480 exam-style questions on AQA A Level Maths 1.11 H: Integration, 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). Each one has a worked solution and a mark scheme showing where the marks go.