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# 大学数学公式集

## 積分

• ${\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}$
• ${\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2}}}+C}$
• ${\displaystyle \int \arctan x\,dx=x\arctan x-{\frac {1}{2}}\ln(1+x^{2})+C}$
• ${\displaystyle \int {\frac {1}{x^{2}+a^{2}}}\,dx={\frac {1}{a}}\arctan {\frac {x}{a}}+C\quad (a\neq 0)}$
• ${\displaystyle \int {\frac {1}{x^{2}-a^{2}}}\,dx={\frac {1}{2a}}\ln \left|{\frac {x-a}{x+a}}\right|+C\quad (a\neq 0)}$
• ${\displaystyle \int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\,dx=\arcsin {\frac {x}{a}}+C\quad (a>0)}$
• ${\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,dx={\frac {1}{2}}\left(x{\sqrt {a^{2}-x^{2}}}+a^{2}\arcsin {\frac {x}{a}}\right)+C\quad (a>0)}$
• ${\displaystyle \int {\frac {1}{\sqrt {x^{2}+A}}}\,dx=\ln \left|x+{\sqrt {x^{2}+A}}\right|+C\quad (A\neq 0)}$
• ${\displaystyle \int {\sqrt {x^{2}+A}}\,dx={\frac {1}{2}}\left(x{\sqrt {x^{2}+A}}+A\ln \left|x+{\sqrt {x^{2}+A}}\right|\right)+C\quad (A\neq 0)}$
• ${\displaystyle \int {\frac {1}{\sin x}}\,dx=\ln \left|\tan {\frac {x}{2}}\right|+C}$
• ${\displaystyle \int {\frac {1}{\cos x}}\,dx={\frac {1}{2}}\ln {\frac {1+\sin x}{1-\sin x}}+C}$
• ${\displaystyle \int {\frac {1}{\tan x}}\,dx=\ln |\sin x|+C}$

## 複素解析

• ${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$