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.
363 exam-style questions on CCEA A Level Maths 3.6 Integration (A-level only), covering 3.6.1 Integration (A-level only), 3.6.2 Integration (A-level only), 3.6.3 Integration (A-level only), 3.6.4 Integration (A-level only), 3.6.5 Integration (A-level only), 3.6.6 Integration (A-level only), and 3.6.7 Integration (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.