Question Number 41108 by mondodotto@gmail.com last updated on 02/Aug/18
$$\boldsymbol{\mathrm{evaluate}}\:\int\boldsymbol{{x}}^{\boldsymbol{{i}}} \boldsymbol{{dx}} \\ $$$$\boldsymbol{\mathrm{where}}\:\boldsymbol{{i}}=\sqrt{−\mathrm{1}} \\ $$
Answered by MJS last updated on 02/Aug/18
$$\int{x}^{{z}} {dx}=\frac{\mathrm{1}}{{z}+\mathrm{1}}{x}^{{z}+\mathrm{1}} \\ $$$$\int{x}^{\mathrm{i}} {dx}=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{i}}{x}^{\mathrm{1}+\mathrm{i}} =\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{i}\right){x}^{\mathrm{1}+\mathrm{i}} \\ $$