Question Number 136067 by bramlexs22 last updated on 18/Mar/21
$$\Lambda\:=\:\int\:{x}^{\mathrm{3}} \:\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{10}} \:{dx}\: \\ $$
Answered by Olaf last updated on 18/Mar/21
$$\Lambda\:=\:\int{x}^{\mathrm{3}} \left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{10}} {dx} \\ $$$$\Lambda\:=\:\int{x}^{\mathrm{3}} \underset{{k}=\mathrm{0}} {\overset{\mathrm{10}} {\sum}}\mathrm{C}_{{k}} ^{\mathrm{10}} {x}^{\mathrm{3}{k}} {dx} \\ $$$$\Lambda\:=\:\int\underset{{k}=\mathrm{0}} {\overset{\mathrm{10}} {\sum}}\mathrm{C}_{{k}} ^{\mathrm{10}} {x}^{\mathrm{3}{k}+\mathrm{3}} {dx} \\ $$$$\Lambda\:=\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{10}} {\sum}}\frac{\mathrm{C}_{{k}} ^{\mathrm{10}} }{\mathrm{3}{k}+\mathrm{4}}{x}^{\mathrm{3}{k}+\mathrm{4}} +\mathrm{C} \\ $$$$\Lambda\:=\:\frac{\mathrm{1}}{\mathrm{34}}{x}^{\mathrm{34}} +\frac{\mathrm{10}}{\mathrm{31}}{x}^{\mathrm{31}} +\frac{\mathrm{45}}{\mathrm{28}}{x}^{\mathrm{28}} +\frac{\mathrm{24}}{\mathrm{5}}{x}^{\mathrm{25}} +\frac{\mathrm{105}}{\mathrm{11}}{x}^{\mathrm{22}} \\ $$$$+\frac{\mathrm{252}}{\mathrm{19}}{x}^{\mathrm{19}} +\frac{\mathrm{105}}{\mathrm{8}}{x}^{\mathrm{16}} +\frac{\mathrm{120}}{\mathrm{13}}{x}^{\mathrm{13}} +\frac{\mathrm{9}}{\mathrm{2}}{x}^{\mathrm{10}} +\frac{\mathrm{10}}{\mathrm{7}}{x}^{\mathrm{7}} \\ $$$$+\frac{\mathrm{1}}{\mathrm{4}}{x}^{\mathrm{4}} +\mathrm{C} \\ $$
Commented by bramlexs22 last updated on 19/Mar/21
$${yes}….{thanks} \\ $$
Answered by Ñï= last updated on 18/Mar/21
$$\Lambda=\int{x}^{\mathrm{3}} \left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{10}} {dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{33}}\int{xd}\left[\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{11}} \right] \\ $$$$=\frac{\mathrm{1}}{\mathrm{33}}\left\{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{11}} {x}−\int\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{11}} {dx}\right\} \\ $$$$=\frac{\mathrm{1}}{\mathrm{33}}\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{11}} {x}−\frac{\mathrm{1}}{\mathrm{33}}\underset{{n}=\mathrm{0}} {\overset{\mathrm{11}} {\sum}}\begin{pmatrix}{\mathrm{11}}\\{{n}}\end{pmatrix}\int{x}^{\mathrm{3}} {dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{33}}\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{11}} {x}−\frac{\mathrm{1}}{\mathrm{132}}\underset{{n}=\mathrm{0}} {\overset{\mathrm{11}} {\sum}}\begin{pmatrix}{\mathrm{11}}\\{{n}}\end{pmatrix}{x}^{\mathrm{4}} +{C} \\ $$