Question Number 145259 by mathdanisur last updated on 03/Jul/21
$$\int\:\frac{\left(\mathrm{3}\sqrt{{x}}+\mathrm{2}\right)^{\mathrm{5}} }{\:\sqrt{{x}}}\:{dx}\:=\:? \\ $$
Answered by Dwaipayan Shikari last updated on 03/Jul/21
$${put}\:\sqrt{{x}}={t}\Rightarrow\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}}{dx}={dt} \\ $$$$=\mathrm{2}.\mathrm{3}^{\mathrm{5}} \int\left({t}+\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{5}} {dt} \\ $$$$=\mathrm{3}^{\mathrm{4}} \left({t}+\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{6}} +{C}=\mathrm{81}\left(\sqrt{{x}}+\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{6}} +{C} \\ $$
Commented by mathdanisur last updated on 03/Jul/21
$${thanks}\:{Ser} \\ $$$${Answer}:\:\frac{\mathrm{1}}{\mathrm{81}}\left(\mathrm{3}\sqrt{{x}}+\mathrm{2}\right)^{\mathrm{6}} \:\boldsymbol{{or}}\:\frac{\mathrm{1}}{\mathrm{9}}\left(\mathrm{3}\sqrt{{x}}+\mathrm{2}\right)^{\mathrm{6}} \:? \\ $$
Commented by Dwaipayan Shikari last updated on 03/Jul/21
$$\frac{\mathrm{1}}{\mathrm{9}}\left(\mathrm{3}\sqrt{{x}}+\mathrm{2}\right)^{\mathrm{6}} +{C} \\ $$
Commented by mathdanisur last updated on 03/Jul/21
$${Ser},\:{but}\:{in}\:{your}\:{solution}\:\mathrm{81}\left(\sqrt{{x}}+\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{6}} \\ $$
Commented by Dwaipayan Shikari last updated on 03/Jul/21
$$\mathrm{81}\left(\sqrt{{x}}+\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{6}} =\frac{\mathrm{3}^{\mathrm{4}} }{\mathrm{3}^{\mathrm{6}} }\left(\mathrm{3}\sqrt{{x}}+\mathrm{2}\right)^{\mathrm{6}} =\frac{\left(\mathrm{3}\sqrt{{x}}+\mathrm{2}\right)^{\mathrm{6}} }{\mathrm{9}} \\ $$
Commented by mathdanisur last updated on 03/Jul/21
$${Thanks}\:{Ser} \\ $$