Determine the indefinite integral ∫ln(x2)x3 dx\int \frac{\ln(x^2)}{x^3} \, \text{d}x∫x3ln(x2)dx.
A mechanical piston is subject to a variable resistive force F(x)F(x)F(x), where xxx is the displacement in metres from the start of the stroke. The force, in Newtons, is given by
F(x)=3+2x2+ln(x2)x3,x≥1 F(x) = \frac{3 + 2x^2 + \ln(x^2)}{x^3}, \quad x \ge 1 F(x)=x33+2x2+ln(x2),x≥1The work done by the force as the piston moves from x=1x = 1x=1 to x=2x = 2x=2 is given by ∫12F(x) dx\int_1^2 F(x) \, \text{d}x∫12F(x)dx.
Using the result from part (a), find the exact work done, writing your answer in the form a+lnba + \ln ba+lnb, where aaa and bbb are constants to be determined.
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.