integral of (x^2-x+6)/((x)(x^2+3))

解

\int \frac{x ^{2} - x + 6}{( x ) ( x ^{2} + 3 )} d x

2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3}) + C

解答ステップ

\int \frac{x ^{2} - x + 6}{x ( x ^{2} + 3 )} d x

以下の部分分数を得る:

\frac{x ^{2} - x + 6}{x ( x ^{2} + 3 )} : \frac{2}{x} + \frac{- x - 1}{x ^{2} + 3}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= 2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3})

定数を解答に追加する

= 2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3}) + C