部分積分法の応用 - xのn次式とxのn次式の積の積分，xのn次式と三角関数の積の積分，xのn次式と指数関数の積の積分

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

イメージ 1

部分積分法の応用で、xのn次式とxのn次式の積の積分、xのn次式と三角関数の積の積分、xのn次式と指数関数の積の積分です。

xのn次式とxのn次式の積の積分

$\displaystyle \int x ( x + 3 )^5 dx = \frac {1}{6} \int x \left\{ ( x + 3 )^6 \right\}' dx = \frac {1}{6} \left\{ x ( x + 3 )^6 - \int x' ( x + 3 )^6 dx \right\} + \frac {C}{2} \\ \displaystyle = \frac {1}{6} \left\{ x ( x + 3 )^6 - \int ( x + 3 )^6 dx \right\} + \frac {C}{2} = \frac {1}{6} \left\{ x ( x + 3 )^6 - \frac {1}{7} ( x + 3 )^7 \right\} + C \\ \displaystyle = \frac {1}{6 \cdot 7} \left\{ 7 x - ( x + 3 ) \right\} ( x + 3 )^6 + C = \frac {1}{6 \cdot 7} ( 6 x - 3 ) ( x + 3 )^6 + C = \frac {3}{6 \cdot 7} ( 2 x - 1 ) ( x + 3 )^6 + C \\ \displaystyle = \frac {1}{14} ( 2 x - 1 ) ( x + 3 )^6 + C$

$\displaystyle \int x \sqrt {x + 1} dx = \int ( x + 1 )^{\frac {1}{2}} \cdot x dx = \frac {2}{3} \int \left\{ ( x + 1 )^{\frac {3}{2}} \right\}' x dx = \frac {2}{3} \left\{ x ( x + 1 )^{\frac {3}{2}} - \int ( x + 1 )^{\frac {3}{2}} ( x )' dx \right\} + \frac {C}{2} \\ \displaystyle = \frac {2}{3} \left\{ x ( x + 1 )^{\frac {3}{2}} - \int ( x + 1 )^{\frac {3}{2}} dx \right\} + \frac {C}{2} = \frac {2}{3} \left\{ x ( x + 1 )^{\frac {3}{2}} - \frac {1}{\frac {5}{2}} ( x + 1 )^{\frac {5}{2}} \right\} + C \\ \displaystyle = \frac {2}{3} \left\{ x ( x + 1 )^{\frac {3}{2}} - \frac {2}{5} ( x + 1 )^{\frac {5}{2}} \right\} + C = \frac {2}{3} ( x + 1 )^{\frac {3}{2}} \left\{ x - \frac {2}{5} ( x + 1 ) \right\} + C \\ \displaystyle = \frac {2}{3} ( x + 1 )^{\frac {3}{2}} \left\{ \frac {3}{5} x - \frac {2}{5} \right\} + C = \frac {2}{15} ( 3 x - 2 ) ( x + 1 ) \sqrt { x + 1 } + C$

$\displaystyle \int \sqrt {x^2 + a} dx = \int ( x )' \sqrt {x^2 + a} dx = x \sqrt {x^2 + a} - \int x ( \sqrt {x^2 + a} )' dx + C \\ \displaystyle = x \sqrt {x^2 + a} - \int x \left\{ ( x^2 + a )^{ \frac {1}{2} } \right\}' dx + C = x \sqrt {x^2 + a} - \int x \cdot \frac {1}{2} \cdot ( x^2 + a )' \cdot ( x^2 + a )^{ - \frac {1}{2} } dx + C \\ \displaystyle = x \sqrt {x^2 + a} - \int x \cdot \frac {1}{2} \cdot 2 x \cdot ( x^2 + a )^{ - \frac {1}{2} } dx + C = x \sqrt {x^2 + a} - \int \frac {x^2}{ \sqrt {x^2 + a} } dx + C\\ \displaystyle = x \sqrt {x^2 + a} - \int \frac {x^2 + a - a}{ \sqrt {x^2 + a} } dx + C = x \sqrt {x^2 + a} - \int { \left( \sqrt {x^2 + a} - \frac {a}{ \sqrt {x^2 + a} } \right) } dx + C \\ \displaystyle = x \sqrt {x^2 + a} - \int \sqrt {x^2 + a} dx + a \int \frac {1}{ \sqrt {x^2 + a} } dx + C \\ \displaystyle \Leftrightarrow 2 \int \sqrt {x^2 + a} dx = x \sqrt {x^2 + a} + a \int \frac {1}{ \sqrt {x^2 + a} } dx + C \\ \displaystyle = x \sqrt {x^2 + a} + a \int \frac { \frac { x + \sqrt {x^2 + a} }{ \sqrt {x^2 + a} } }{ x + \sqrt {x^2 + a} } dx + C = x \sqrt {x^2 + a} + a \int \frac { ( x + \sqrt {x^2 + a} )' }{ x + \sqrt {x^2 + a} } dx + C \\ \displaystyle = x \sqrt {x^2 + a} + a \log \left| x + \sqrt {x^2 + a} \right| + 2 C \\ \displaystyle \Leftrightarrow \int \sqrt {x^2 + a} dx = \frac {1}{2} ( x \sqrt {x^2 + a} + a \log \left| x + \sqrt {x^2 + a} \right| ) + C$

xのn次式と三角関数の積の積分

$\displaystyle \int x \sin x dx = - \int x ( \cos x )' dx = - x \cos x + \int ( x )' \cos x dx + \frac {C}{2} = - x \cos x + \int \cos x dx + \frac {C}{2} \\ \displaystyle = - x \cos x + \sin x + C$

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

$\displaystyle \int ( x - 2 ) \cos x dx$

$\displaystyle \int \frac { x \cos x - \sin x }{x^2} x dx$

$\displaystyle \int \frac { x \sin x + \cos x }{x^2} x dx$

$\displaystyle \int \frac { x }{\cos^2 x} x dx$

$\displaystyle \int x \tan^2 x dx$

$\displaystyle \int x \tan^4 x dx$

xのn次式と指数関数の積の積分

$\displaystyle \int x^3 e^x dx$

$\displaystyle \int \frac {x^2}{e^x} dx$

$\displaystyle \int ( x^2 - x + 2 ) e^x dx$

$\displaystyle \int x^3 e^{2 x} dx$

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

$\displaystyle \int x^5 e^{x^3} dx = \frac{1}{3} \int x^3 \cdot 3 x^{2} e^{x^{3}} dx = \frac{1}{3} \int x^3 \cdot \left( x^3 \right) ' e^{x^{3}} dx = \frac{1}{3} \int x^3 \cdot \left( e^{x^{3}} \right) ' dx \\ \displaystyle = \frac{1}{3} \left\{ x^3 e^{x^3} - \int \left( x^3 \right) ' e^{x^3} dx \right\} + \frac{C}{2} = \frac{1}{3} \left\{ x^3 e^{x^3} - \int \left( e^{x^3} \right) ' dx \right\} + \frac{C}{2} = \frac{1}{3} \left( x^{3} e^{x^{3}} - e^{x^{3}} \right) + C \\ \displaystyle = \frac{1}{3} \left( x^3 - 1 \right) e^{x^{3}} + C = \frac{1}{3} ( x - 1 ) \left( x^2 + x + 1 \right) e^{x^{3}} + C$

一般化すると次のように表せる。

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