integral of (4-x)/((x+2)(x^2+2))

解

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

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

解答ステップ

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

以下の部分分数を得る:

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

= \int \frac{1}{x + 2} + \frac{- x + 1}{x ^{2} + 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

= \int \frac{1}{x + 2} d x + \int \frac{- x + 1}{x ^{2} + 2} d x

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

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

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

定数を解答に追加する

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