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人気のある微分積分 >

integral of (50)/((x-1)(x^2+49))

解

\int \frac{50}{( x - 1 ) ( x ^{2} + 49 )} d x

解

ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}) + C

解答ステップ

\int \frac{50}{( x - 1 ) ( x ^{2} + 49 )} d x

定数を除く:

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

= 50 \cdot \int \frac{1}{( x - 1 ) ( x ^{2} + 49 )} d x

以下の部分分数を得る:

\frac{1}{( x - 1 ) ( x ^{2} + 49 )} : \frac{1}{50 ( x - 1 )} + \frac{- x - 1}{50 ( x ^{2} + 49 )}

= 50 \cdot \int \frac{1}{50 ( x - 1 )} + \frac{- x - 1}{50 ( x ^{2} + 49 )} 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

= 50 (\int \frac{1}{50 ( x - 1 )} d x + \int \frac{- x - 1}{50 ( x ^{2} + 49 )} d x)

\int \frac{1}{50 ( x - 1 )} d x = \frac{1}{50} ln ∣ x - 1 ∣

\int \frac{- x - 1}{50 ( x ^{2} + 49 )} d x = \frac{1}{50} (- \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}))

= 50 (\frac{1}{50} ln ∣ x - 1 ∣ + \frac{1}{50} (- \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7})))

簡素化

50 (\frac{1}{50} ln ∣ x - 1 ∣ + \frac{1}{50} (- \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}))) : ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7})

= ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7})

定数を解答に追加する

= ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}) + C

グラフ

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