Question Number 185951 by manxsol last updated on 30/Jan/23
$$\int{x}^{\mathrm{3}} {ln}\left({x}+\mathrm{4}\right){dx} \\ $$
Answered by SEKRET last updated on 30/Jan/23
$$\:\int\:\boldsymbol{\mathrm{x}}^{\mathrm{3}} \centerdot\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}+\mathrm{4}\right)\:\boldsymbol{\mathrm{dx}}= \\ $$$$\:\:=\frac{\boldsymbol{\mathrm{x}}^{\mathrm{4}} }{\mathrm{4}}\centerdot\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}+\mathrm{4}\right)\:−\:\int\:\frac{\boldsymbol{\mathrm{x}}^{\mathrm{4}} }{\mathrm{4}}\centerdot\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}+\mathrm{4}}\boldsymbol{\mathrm{dx}} \\ $$$$=\:\frac{\boldsymbol{\mathrm{x}}^{\mathrm{4}} }{\mathrm{4}}\centerdot\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}+\mathrm{4}\right)−\frac{\mathrm{1}}{\mathrm{4}}\centerdot\int\left(\boldsymbol{\mathrm{x}}^{\mathrm{3}} −\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{16}\boldsymbol{\mathrm{x}}+\frac{\mathrm{256}}{\boldsymbol{\mathrm{x}}+\mathrm{4}}−\mathrm{64}\right)\boldsymbol{\mathrm{dx}} \\ $$$$=\frac{\boldsymbol{\mathrm{x}}^{\mathrm{4}} }{\mathrm{4}}\centerdot\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}−\mathrm{4}\right)−\frac{\mathrm{1}}{\mathrm{4}}\centerdot\left(\frac{\boldsymbol{\mathrm{x}}^{\mathrm{4}} }{\mathrm{4}}−\frac{\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{3}} }{\mathrm{3}}+\mathrm{8}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{256}\centerdot\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}+\mathrm{4}\right)−\mathrm{64}\boldsymbol{\mathrm{x}}\right)+\boldsymbol{\mathrm{C}} \\ $$