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人気のある微分積分 >

integral of (48x^2)/((x-24)(x+8)^2)

解

\int \frac{48 x ^{2}}{( x - 24 ) ( x + 8 ) ^{2}} d x

解

48 (\frac{9}{16} ln ∣ x - 24 ∣ + \frac{7}{16} ln ∣ x + 8 ∣ + \frac{2}{x + 8}) + C

解答ステップ

\int \frac{48 x ^{2}}{( x - 24 ) ( x + 8 ) ^{2}} d x

定数を除く:

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

= 48 \cdot \int \frac{x ^{2}}{( x - 24 ) ( x + 8 ) ^{2}} d x

以下の部分分数を得る:

\frac{x ^{2}}{( x - 24 ) ( x + 8 ) ^{2}} : \frac{9}{16 ( x - 24 )} + \frac{7}{16 ( x + 8 )} - \frac{2}{( x + 8 ) ^{2}}

= 48 \cdot \int \frac{9}{16 ( x - 24 )} + \frac{7}{16 ( x + 8 )} - \frac{2}{( x + 8 ) ^{2}} d x

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

= 48 (\int \frac{9}{16 ( x - 24 )} d x + \int \frac{7}{16 ( x + 8 )} d x - \int \frac{2}{( x + 8 ) ^{2}} d x)

\int \frac{9}{16 ( x - 24 )} d x = \frac{9}{16} ln ∣ x - 24 ∣

\int \frac{7}{16 ( x + 8 )} d x = \frac{7}{16} ln ∣ x + 8 ∣

\int \frac{2}{( x + 8 ) ^{2}} d x = - \frac{2}{x + 8}

= 48 (\frac{9}{16} ln ∣ x - 24 ∣ + \frac{7}{16} ln ∣ x + 8 ∣ - (- \frac{2}{x + 8}))

簡素化

= 48 (\frac{9}{16} ln ∣ x - 24 ∣ + \frac{7}{16} ln ∣ x + 8 ∣ + \frac{2}{x + 8})

定数を解答に追加する

= 48 (\frac{9}{16} ln ∣ x - 24 ∣ + \frac{7}{16} ln ∣ x + 8 ∣ + \frac{2}{x + 8}) + C

グラフ

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