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$\{ f ( x ) \}^n f ' ( x )$ 型の積分の応用 - $f ( x )$ が $x$ の $n$ 次式の積分

イメージ 1

$\{ f ( x ) \}^n f ' ( x )$ 型の積分で、 $f ( x )$ が $x$ の $n$ 次式の積分です。

$\displaystyle \int ( 2 x - 3 )^4 dx = \frac{1}{2} \int 2 \cdot ( 2 x - 3 )^4 dx = \frac{1}{2} \int ( 2 x - 3 ) ' ( 2 x - 3 )^4 dx = \frac{1}{2 \cdot 5} ( 2 x - 3 )^5 + C = \frac{1}{10} ( 2 x - 3 )^5 + C$

$\displaystyle \int x^2 \left( x^3 + 2 \right)^5 dx = \frac{ 1 }{ 3 } \int 3 x^2 \left( x^3 + 2 \right)^5 dx = \frac{ 1 }{ 3 } \int \left( x^3 + 2 \right) ' \left( x^3 + 2 \right)^5 dx = \frac{ 1 }{ 3 \cdot 6 } \left( x^3 + 2 \right)^6 + C \\ \displaystyle = \frac{ 1 }{ 18 } \left( x^3 + 2 \right)^6 + C$

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

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

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

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

$\displaystyle \int ( x - 1 ) \sqrt{ x + 3 } dx = \int \left\{ ( x + 3 ) - 4 \right\} \sqrt{ x + 3 } dx = \int ( x + 3 ) \sqrt{ x + 3 } dx - 4 \int \sqrt{ x + 3 } dx \\ \displaystyle = \int ( x + 3 )^{\frac{ 3 }{ 2 }} dx - 4 \int ( x + 3 )^{\frac{ 1 }{ 2 }} dx = \frac{ ( x + 3 )^{\frac{ 5 }{ 2 }} }{ \dfrac{ 5 }{ 2 } } - 4 \cdot \frac{ ( x + 3 )^{\frac{ 3 }{ 2 }} }{ \dfrac{ 3 }{ 2 } } + C \\ \displaystyle = \frac{ 2 }{ 5 } ( x + 3 )^2 \sqrt{ x + 3 } - \frac{ 8 }{ 3 } ( x + 3 ) \sqrt{ x + 3 } + C = \left\{ \frac{ 2 }{ 5 } ( x + 3 ) - \frac{ 8 }{ 3 } \right\} ( x + 3 ) \sqrt{ x + 3 } + C \\ \displaystyle = \frac{ 2 }{ 15 } ( 3 x - 11 ) ( x + 3 ) \sqrt{ x + 3 }$

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

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