integral of (2x-x^2-1)/(x^3+x)

解

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

2 arctan (x) - ln ∣ x ∣ + C

解答ステップ

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

拡張

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

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

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

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

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

簡素化

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

= 2 arctan (x) - ln ∣ x ∣

定数を解答に追加する

= 2 arctan (x) - ln ∣ x ∣ + C