# x-5-x-3-1-1-3-dx-

Question Number 11149 by suci last updated on 14/Mar/17
$$\int{x}^{\mathrm{5}} \:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx}=…??? \\$$
Answered by ajfour last updated on 14/Mar/17
$$\:\frac{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{7}/\mathrm{3}} }{\mathrm{7}}−\frac{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{4}/\mathrm{3}} }{\mathrm{4}}\:+{C}\: \\$$
Commented by ajfour last updated on 14/Mar/17
$${let}\:\:\:{x}^{\mathrm{3}} +\mathrm{1}={t} \\$$$$\mathrm{3}{x}^{\mathrm{2}} {dx}={dt} \\$$$${so},\:\int{x}^{\mathrm{5}} \left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} {dx}\: \\$$$$=\:\frac{\mathrm{1}}{\mathrm{3}}\int{x}^{\mathrm{3}} \left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} \mathrm{3}{x}^{\mathrm{2}} {dx} \\$$$$=\frac{\mathrm{1}}{\mathrm{3}}\int\left({t}−\mathrm{1}\right){t}^{\mathrm{1}/\mathrm{3}} {dt}=\frac{\mathrm{1}}{\mathrm{3}}\int\left({t}^{\mathrm{4}/\mathrm{3}} −{t}^{\mathrm{1}/\mathrm{3}} \right){dt} \\$$$$=\:\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{t}^{\mathrm{7}/\mathrm{3}} }{\mathrm{7}/\mathrm{3}}\right)−\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{t}^{\mathrm{4}/\mathrm{3}} }{\mathrm{4}/\mathrm{3}}\right)\:+{C} \\$$$${hence}\:{the}\:{answer}. \\$$
Answered by Mechas88 last updated on 17/Mar/17
$$\\$$$${u}=\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} \\$$$${u}^{\mathrm{3}} ={x}^{\mathrm{3}} +\mathrm{1} \\$$$${x}^{\mathrm{3}} ={u}^{\mathrm{3}} −\mathrm{1} \\$$$${du}=\frac{\mathrm{1}}{\mathrm{3}}\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{−\mathrm{2}/\mathrm{3}} \left(\mathrm{3}{x}^{\mathrm{2}} \right){dx}={x}^{\mathrm{2}} \left({x}^{\mathrm{3}} +\mathrm{1}\right)^{−\mathrm{2}/\mathrm{3}} {dx} \\$$$${dx}={x}^{−\mathrm{2}} \left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{2}/\mathrm{3}} {du} \\$$$$\\$$$$\int{x}^{\mathrm{5}} {x}^{−\mathrm{2}} {udu}=\int\left({u}^{\mathrm{3}} −\mathrm{1}\right){udu}= \\$$$$\int{u}^{\mathrm{4}} {du}−\int{udu}= \\$$$$\frac{{u}^{\mathrm{5}} }{\mathrm{5}}\:−\:\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:+\:{C}= \\$$$$\frac{\sqrt[{\mathrm{3}}]{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{5}} }\:}{\mathrm{5}}\:−\frac{\sqrt[{\mathrm{3}}]{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{2}} \:\:\:\:}}{\mathrm{2}}\:\:+\:\:{C}\:\:\ast\ast\ast{Rta} \\$$$$\\$$$$\\$$$$\\$$