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≥1F(x) = \frac{3 + 2x^2 + \ln(x^2)}{x^3}, \quad x \ge 1F(x)=x33+2x2+ln(x2),x≥1 The 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.
Practise Edexcel A Level Maths 11.6 Integration by Parts with exam-style questions for A Level Maths. 52 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.