基本関数の不定積分の公式

カテゴリー[ 昆虫| 田園| 街| 数学・幾何学| 寺院| 城| 祭り| 鉄道| 海| 風力発電]

基本関数の積分の公式です。基本的に微分の逆を考えていきます。
x^n, 1/x, cos x, sin x, 1/(cos x)^2, 1/(sin x)^2, e^x, a^x。

基本関数の不定 積分の公式

微分の公式より、次のような積分公式が導出できる。

$\displaystyle \int {x^n} dx = {\frac {1}{n + 1} } x^{n + 1} + C$ $( n \neq - 1 )$

$\displaystyle \int {\frac {1}{x} } dx = \log | x | + C$

$\displaystyle \int {\cos x } dx = \sin x + C$

$\displaystyle \int {\sin x } dx = - \cos x + C$

$\displaystyle \int {\frac {1}{\cos ^2 x}} dx = \int {( \tan ^2 x + 1 )} dx = \tan x + C$

$\displaystyle \int {\frac {1}{\sin ^2 x}} dx = \int {\left( \frac {1}{\tan ^2 x} + 1 \right)} dx = - \frac {1}{\tan x} + C$

$\displaystyle \int {e^x} dx = e^x + C$

$\displaystyle \int {a^x} dx = \frac {a^x} {\log a} + C$

応用

$\displaystyle \int {\frac {1} {x^2}} dx = \int {x^{- 2}} dx = - x^{- 1} + C = - \frac {1} {x} + C$

$\displaystyle \int {\sqrt x} dx = \int {x^{\frac {1} {2}}} dx = \frac {x^{\frac {3} {2}}} {\frac {3} {2}} + C = {\frac {2} {3}} x \sqrt x + C$

$\displaystyle \int \frac {1}{\sqrt {x}} dx = \int x^{ - \frac {1}{2} } dx = \frac {x^{\frac {1}{2}}}{\frac {1}{2}} + C = 2 \sqrt {x} + C$

$\displaystyle \int \frac {1}{\sqrt {x} + \sqrt { x + 1 } } dx = \int \frac {\sqrt {x} - \sqrt { x + 1 }}{\left( \sqrt {x} + \sqrt { x + 1 } \right) \left( \sqrt {x} - \sqrt { x + 1 } \right)} dx = \int \frac {\sqrt {x} - \sqrt { x + 1 }}{ x - ( x + 1 ) } dx \\ \displaystyle = - \int \left(\sqrt {x} - \sqrt { x + 1 } \right) dx = \int \left( \sqrt { x + 1 } - \sqrt {x} \right) dx = \int \sqrt { x + 1 } dx - \int \sqrt { x } dx \\ \displaystyle = \int ( x + 1 )^{\frac {1}{2}} dx - \int x^{\frac {1}{2}} dx = \frac {( x + 1 )^{\frac {3}{2}}}{\frac {3}{2}} - \frac {x^{\frac {1}{2}}}{\frac {3}{2}} + C = \frac {2}{3} \left\{ ( x + 1 ) \sqrt { x + 1 } - x \sqrt {x} \right\} + C$

$\displaystyle \int \sin ^2 x dx = \int \frac {1 - \cos 2 x}{2} dx = \frac {1}{2} \int ( 1 - \cos 2 x ) dx = \frac {1}{2} \left( \int dx - \int \cos 2 x dx \right) \\ \displaystyle = \frac {1}{2} \left( x - \frac { \sin 2 x }{2} \right) + C = \frac {1}{2} \left( x - \sin x \cos x \right) + C$

$\displaystyle \int \cos ^2 x dx = \int \frac {1 - \cos 2 x}{2} dx = \frac {1}{2} \int ( 1 + \cos 2 x ) dx = \frac {1}{2} \left( \int dx + \int \cos 2 x dx \right) \\ \displaystyle = \frac {1}{2} \left( x + \frac { \sin 2 x }{2} \right) + C = \frac {1}{2} \left( x + \sin x \cos x \right) + C$

$\displaystyle \int \tan ^2 x dx = \int \frac {\sin ^2 x}{\cos ^2 x} dx = \int \frac {1 - \cos ^2 x}{\cos ^2 x} dx = \int \left( \frac {1}{\cos ^2 x} - 1 \right) dx = \int \frac {1}{\cos ^2 x} dx - \int dx \\ \displaystyle = \tan x - x + C$

$\displaystyle \int \sin x \cos x dx = \frac {1}{2} \int \sin 2 x dx = \frac {1}{4} \cos 2 x + \left( C + \frac {1}{4} \right) = \frac {1}{4} \left( 1 - 2 \sin ^2 x \right) + \left( C + \frac {1}{4} \right) \\ \displaystyle = \frac {1}{2} \sin ^2 x + C$

$\displaystyle \int \frac { e^x - e^{ - x } }{2} dx = \frac {1}{2} \int \left( e^x - e^{ - x } \right) dx = \frac {1}{2} \left( \int e^x dx - \int e^{ - x } dx \right) = \frac {1}{2} \left( e^x + e^{ - x } \right) + C = \frac {e^x + e^{ - x } }{2} + C$

$\displaystyle \int \frac { e^x + e^{ - x } }{2} dx = \frac {1}{2} \int \left( e^x + e^{ - x } \right) dx = \frac {1}{2} \left( \int e^x dx + \int e^{ - x } dx \right) = \frac {1}{2} \left( e^x - e^{ - x } \right) + C = \frac {e^x - e^{ - x } }{2} + C$