Question Number 185959 by Michaelfaraday last updated on 30/Jan/23
$${solve}: \\ $$$$\int{x}^{{Inx}} {dx} \\ $$
Answered by Frix last updated on 30/Jan/23
$$\int{x}^{\mathrm{ln}\:{x}} {dx}\:\overset{{t}=\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{ln}\:{x}} {=}\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{\mathrm{e}}}\int\mathrm{e}^{{t}^{\mathrm{2}} } {dt}=\frac{\sqrt{\pi}}{\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{e}}}\mathrm{erfi}\:\left({t}\right)\:= \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{e}}}\mathrm{erfi}\:\left(\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{ln}\:{x}\right)\:+{C} \\ $$
Commented by Michaelfaraday last updated on 30/Jan/23
$${thanks}\:{sir} \\ $$