integral of (x^2+4x+10)/(x^3+2x^2+5x)

解

\int \frac{x ^{2} + 4 x + 10}{x ^{3} + 2 x ^{2} + 5 x} d x

2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 2 x + 5 + \frac{1}{2} arctan (\frac{1}{2} (x + 1)) + C

解答ステップ

\int \frac{x ^{2} + 4 x + 10}{x ^{3} + 2 x ^{2} + 5 x} d x

以下の部分分数を得る:

\frac{x ^{2} + 4 x + 10}{x ^{3} + 2 x ^{2} + 5 x} : \frac{2}{x} - \frac{x}{x ^{2} + 2 x + 5}

= \int \frac{2}{x} - \frac{x}{x ^{2} + 2 x + 5} 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}{x ^{2} + 2 x + 5} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{x}{x ^{2} + 2 x + 5} d x = \frac{1}{2} ln x^{2} + 2 x + 5 - \frac{1}{2} arctan (\frac{1}{2} (x + 1))

= 2 ln ∣ x ∣ - (\frac{1}{2} ln x^{2} + 2 x + 5 - \frac{1}{2} arctan (\frac{1}{2} (x + 1)))

簡素化

= 2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 2 x + 5 + \frac{1}{2} arctan (\frac{1}{2} (x + 1))

定数を解答に追加する

= 2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 2 x + 5 + \frac{1}{2} arctan (\frac{1}{2} (x + 1)) + C