iのi乗
\[
\Im\left(i^{i}\right)=0
\]
カテゴリー: 問題
log₂3とlog₃5の大小比較
\[
\log_{2}3\lesseqgtr\log_{3}5
\]
eのπ乗とπのe乗の大小比較
\[
e^{\pi}\lesseqgtr\pi^{e}
\]
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
\[
\int\sqrt{\tan x}dx=\frac{\sqrt{2}}{4}\log\left(\tan x-\sqrt{2\tan x}+1\right)-\frac{\sqrt{2}}{4}\log\left(\tan x+\sqrt{2\tan x}+1\right)+\frac{\sqrt{2}}{2}\tan^{\circ}\left(\sqrt{2\tan x}-1\right)+\frac{\sqrt{2}}{2}\tan^{\circ}\left(\sqrt{2\tan x}+1\right)+C
\]
対数のルート積分
\[
\int\log^{\frac{1}{2}}xdx=x\log^{\frac{1}{2}}x-\frac{\sqrt{\pi}}{2}erfi\left(\log^{\frac{1}{2}}x\right)+C
\]
分母に2乗根と3乗根の積分
\[
\int\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}dx=2x^{\frac{1}{2}}-3x^{\frac{1}{3}}+6x^{\frac{1}{6}}-6\log\left(1+x^{\frac{1}{6}}\right)
\]
分母に(1+x²)²を含む積分
\[
\int\frac{1}{\left(1+x^{2}\right)^{2}}dx=\frac{1}{2}\tan^{\circ}x+\frac{x}{2\left(1+x^{2}\right)}+C
\]