Mathematics · Further integration · 5 marks
Find ∫ x² ln x dx. [5]
Mark scheme
- By parts with u = ln x and dv/dx = x²
- du/dx = 1/x and v = x³/3
- = (x³/3) ln x − ∫ (x³/3)(1/x) dx
- = (x³/3) ln x − ∫ x²/3 dx
- = (x³/3) ln x − x³/9 + c
Take u = ln x even though it is not the first factor: it is the part that gets simpler when differentiated, and ln x is one of the few things you cannot integrate directly at this level.