iのi乗
\(i\)の\(i\)乗
\[
\Im\left(i^{i}\right)=0
\]
\(\Im\left(z\right)\)は\(z\)の虚部。
\begin{align*} \Im\left(i^{i}\right) & =\Im\left(\left(e^{\frac{\pi}{2}i}\right)^{i}\right)\\ & =\Im\left(e^{-\frac{\pi}{2}}\right)\\ & =0 \end{align*}
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- tanの立方根の積分\[ \int\sqrt[3]{\tan x}dx=\frac{1}{4}\log\left(\tan^{\frac{4}{3}}x-\tan^{\frac{2}{3}}x+1\right)+\frac{\sqrt{3}}{2}\tan^{\circ}\left(\frac{2\tan^{\frac{2}{3}}x-1}{\sqrt{3}}\right)-\frac{1}{2}\log\left(\tan^{\frac{2}{3}}x+1\right)+C \]
- tanの平方根の積分\[ \int\sqrt{\tan x}dx=\frac{\sqrt{2}}{4}\log\left(\tan x-\sqrt{2\tan x}+1\right)-\frac{\sqrt{2}}{4}\log\left(\tan x+\sqrt{2\tan x}+1\right)+\frac{\sqrt{2}}{2}\tan^{\circ}\left(\sqrt{2\tan x}-1\right)+\frac{\sqrt{2}}{2}\tan^{\circ}\left(\sqrt{2\tan x}+1\right)+C \]
- eのπ乗とπのe乗の大小比較\[ e^{\pi}\lesseqgtr\pi^{e} \]
- 分母に(1+x²)²を含む積分\[ \int\frac{1}{\left(1+x^{2}\right)^{2}}dx=\frac{1}{2}\tan^{\circ}x+\frac{x}{2\left(1+x^{2}\right)}+C \]