数学の基礎と公式-5
数学の基礎と公式
amazon kindle版を出版しました。
(1)$\int(x)dx={\dfrac{x^2}{2}},\int(x^2)dx={\dfrac{x^3}{3}}, \cdots $
$\int(x^n)dx={\dfrac{x^{n+1}}{n+1}}(n \neq -1)$
(2)$\displaystyle \int\dfrac{1}{x} dx=\log | x |$
(3)$\int e^x dx=e^x$
(4)$\int y^x \log y dx=y^x$
(5)$\int x \log x dx= {\dfrac{x^2\,\log x}{2}}-{\dfrac{x^2}{4}}$
(6)$\int \log x dx=x \log x -x$
(7)$\int \log (x+y) dx=(x+y) \log (x+y) -(x+y)$
(8)$\int \log x^2 dx=2(x \log x -x)$
(10)$\int \log (x+y)^2 dx=2\{(x+y) \log (x+y) -(x+y)\}$
(11)$\int (\log x)^2 dx=x \{(\log x)^2 - 2 \log x + 2\}$
(12)$\int \log (x+y)^2 dx=(x+y) \{(\log (x+y))^2 - 2 \log (x+y) + 2\}$
(13)$\int (\log x^2)^2 dx=4 x \{(\log x)^2 - 2 \log x + 2\}$
(14)$\displaystyle \int\dfrac{\log x}{x} dx=\dfrac{(\log x)^2}{2}$
(15)$\displaystyle \int\dfrac{\log x}{x^2} dx=-\dfrac{\log x}{x}-\dfrac{1}{x}$
(16)$\displaystyle \int\dfrac{1}{x \log x} dx=\log (\log x)$
(17)$\displaystyle \int\dfrac{\log (\log x)}{x} dx=\log x\,\log (\log x)-\log x$
(18)$\displaystyle \int\dfrac{1}{x (\log x+1)^2} dx=\dfrac{1}{\log x+1}$
(19)$\displaystyle \int\ \log (x^2+y^2) dx=x \log (x^2+y^2)-2(x-y \tan \dfrac{x}{y} )$
(20)$\displaystyle \int\ \log (x^2-y^2) dx$
$= x \log (x^2-y^2)+y \log(x+y)-y \log(x-y)-2x$
$= (x+y) \log (x+y)+(x-y) \log(x-y)-2x$
(21)$\displaystyle \int \dfrac{\log(\sin^2 x)}{\tan x} dx=\{\log(\sin x)\}^2$
(22)$\displaystyle \int \tan x \log(\cos x) dx=-\dfrac{(\log(\cos x))^2}{2}$
(23)$\displaystyle \int \dfrac{ \log(\tan x)}{\tan x (\cos x)^2} dx=-\dfrac{(\log(\tan x))^2}{2}$
※$\log x$の$x$の箇所には絶対値がつきます。$\log |x|$
