Additional Mathematics 0606 / Calculus — Integration / 2.352.35: Product rule and logarithmic integration0606/22/M/J/17 — Question 5 · 5 marksMark as done · Save for later · Show all solutions(i)Show that ddx[0.4x5(0.2−ln5x)]=kx4ln5x\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl[0.4x^{5}(0.2 - \ln 5x)\bigr] = kx^{4}\ln 5xdxd[0.4x5(0.2−ln5x)]=kx4ln5x, where kkk is an integer to be found.[2]▸ Answer(ii)Express ln125x3\ln 125x^{3}ln125x3 in terms of ln5x\ln 5xln5x.[1]▸ Answer(iii)Hence find ∫(x4ln125x3) dx\displaystyle\int (x^{4}\ln 125x^{3})\,\mathrm{d}x∫(x4ln125x3)dx.[2]▸ Answer▸ Official mark scheme← 2.3: Finding the equation of a curve from its gradient2.36: Area between a cubic and a horizontal line →
