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

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