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人気のある微分積分 >

integral of (x^2)/((3+4x-4x^2)^{3/2)}

解

\int \frac{x ^{2}}{( 3 + 4 x - 4 x ^{2} ) ^{\frac{3}{2}}} d x

解

\frac{10 x - 4 arcsin ( x - \frac{1}{2} ) ( - 4 x ^{2} + 4 x + 3 ) ^{\frac{1}{2}} + 3}{32 ( - 4 x ^{2} + 4 x + 3 ) ^{\frac{1}{2}}} + C

解答ステップ

\int \frac{x ^{2}}{( 3 + 4 x - 4 x ^{2} ) ^{\frac{3}{2}}} d x

平方完成する

3 + 4 x - 4 x^{2} : - 4 (x - \frac{1}{2})^{2} + 4

= \int \frac{x ^{2}}{( - 4 ( x - \frac{1}{2} ) ^{2} + 4 ) ^{\frac{3}{2}}} d x

u置換積分法を適用する

= \int \frac{( 2 u + 1 ) ^{2}}{32 ( - u ^{2} + 1 ) ^{\frac{3}{2}}} d u

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{32} \cdot \int \frac{( 2 u + 1 ) ^{2}}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} d u

拡張

\frac{( 2 u + 1 ) ^{2}}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} : \frac{4 u ^{2}}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} + \frac{4 u}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} + \frac{1}{( - u ^{2} + 1 ) ^{\frac{3}{2}}}

= \frac{1}{32} \cdot \int \frac{4 u ^{2}}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} + \frac{4 u}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} + \frac{1}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} d u

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{32} (\int \frac{4 u ^{2}}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} d u + \int \frac{4 u}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} d u + \int \frac{1}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} d u)

\int \frac{4 u ^{2}}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} d u = 4 (- arcsin (u) + \frac{u}{( 1 - u ^{2} ) ^{\frac{1}{2}}})

\int \frac{4 u}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} d u = \frac{4}{( - u ^{2} + 1 ) ^{\frac{1}{2}}}

\int \frac{1}{( - u ^{2} + 1 ) ^{\frac{3}{2}}} d u = \frac{u}{( 1 - u ^{2} ) ^{\frac{1}{2}}}

= \frac{1}{32} (4 (- arcsin (u) + \frac{u}{( 1 - u ^{2} ) ^{\frac{1}{2}}}) + \frac{4}{( - u ^{2} + 1 ) ^{\frac{1}{2}}} + \frac{u}{( 1 - u ^{2} ) ^{\frac{1}{2}}})

代用を戻す

u = x - \frac{1}{2}

= \frac{1}{32} 4 - arcsin (x - \frac{1}{2}) + \frac{x - \frac{1}{2}}{( 1 - ( x - \frac{1}{2} ) ^{2} ) ^{\frac{1}{2}}} + \frac{4}{( - ( x - \frac{1}{2} ) ^{2} + 1 ) ^{\frac{1}{2}}} + \frac{x - \frac{1}{2}}{( 1 - ( x - \frac{1}{2} ) ^{2} ) ^{\frac{1}{2}}}

簡素化

\frac{1}{32} 4 - arcsin (x - \frac{1}{2}) + \frac{x - \frac{1}{2}}{( 1 - ( x - \frac{1}{2} ) ^{2} ) ^{\frac{1}{2}}} + \frac{4}{( - ( x - \frac{1}{2} ) ^{2} + 1 ) ^{\frac{1}{2}}} + \frac{x - \frac{1}{2}}{( 1 - ( x - \frac{1}{2} ) ^{2} ) ^{\frac{1}{2}}} : \frac{10 x - 4 arcsin ( x - \frac{1}{2} ) ( - 4 x ^{2} + 4 x + 3 ) ^{\frac{1}{2}} + 3}{32 ( - 4 x ^{2} + 4 x + 3 ) ^{\frac{1}{2}}}

= \frac{10 x - 4 arcsin ( x - \frac{1}{2} ) ( - 4 x ^{2} + 4 x + 3 ) ^{\frac{1}{2}} + 3}{32 ( - 4 x ^{2} + 4 x + 3 ) ^{\frac{1}{2}}}

定数を解答に追加する

= \frac{10 x - 4 arcsin ( x - \frac{1}{2} ) ( - 4 x ^{2} + 4 x + 3 ) ^{\frac{1}{2}} + 3}{32 ( - 4 x ^{2} + 4 x + 3 ) ^{\frac{1}{2}}} + C

グラフ

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