$\displaystyle f(x)$ | $\displaystyle \int f(x)dx$ | 자주 나오는 다른 표현 | 증명 |
$\displaystyle \sin x$ | $\displaystyle -\cos x$ | | sin_x_적분_증명 |
$\displaystyle \cos x$ | $\displaystyle \sin x$ | | cos_x_적분_증명 |
$\displaystyle \tan x$ | $\displaystyle \ln|\sec x|$ | $\displaystyle -\ln|\cos x|$ | tan_x_적분_증명 |
$\displaystyle \csc x$ | $\displaystyle \ln|\csc x-\cot x|$ | $\displaystyle -\ln|\csc x+\cot x|$ | csc_x_적분_증명 |
$\displaystyle \sec x$ | $\displaystyle \ln|\sec x+\tan x|$ | | sec_x_적분_증명 |
$\displaystyle \cot x$ | $\displaystyle \ln|\sin x|$ | $\displaystyle -\ln|\csc x|$ | cot_x_적분_증명 |
$\displaystyle \sin^{-1}x$ |
$\displaystyle \cos^{-1}x$ |
$\displaystyle \tan^{-1}x$ |
$\displaystyle \csc^{-1}x$ |
$\displaystyle \sec^{-1}x$ |
$\displaystyle \cot^{-1}x$ |
$\displaystyle \sinh x$ | $\displaystyle \cosh x$ |
$\displaystyle \cosh x$ | $\displaystyle \sinh x$ |
$\displaystyle \tanh x$ | $\displaystyle \ln(\cosh x)$ | 이하 네개 wpko에서 왔음,CHK |
$\displaystyle \operatorname{csch} x$ | $\displaystyle \ln\left|\tanh\frac{x}{2}\right|$ |
$\displaystyle \operatorname{sech} x$ | $\displaystyle \tan^{-1}(\sinh x)$ |
$\displaystyle \coth x$ | $\displaystyle \ln|\sinh x|$ |
$\displaystyle \sinh^{-1} x$ | $\displaystyle x\sinh^{-1}x-\sqrt{x^2+1}$ | | arcsinh_x_적분_증명 |
$\displaystyle \cosh^{-1} x$ | $\displaystyle x\cosh^{-1}x-\sqrt{x^2-1}$ | | arccosh_x_적분_증명 |
$\displaystyle \tanh^{-1} x$ | $\displaystyle x\tanh^{-1}x+\frac12\ln(1-x^2)+C$ | | arctanh_x_적분_증명 |
$\displaystyle \operatorname{csch}^{-1} x$ |
$\displaystyle \operatorname{sech}^{-1} x$ |
$\displaystyle \coth^{-1} x$ |