integral of (-9x^2+2)/(x^3+9x)

解

\int \frac{- 9 x ^{2} + 2}{x ^{3} + 9 x} d x

- \frac{83}{18} ln x^{2} + 9 + \frac{2}{9} ln ∣ x ∣ + C

解答ステップ

\int \frac{- 9 x ^{2} + 2}{x ^{3} + 9 x} d x

拡張

\frac{- 9 x ^{2} + 2}{x ^{3} + 9 x} : - \frac{9 x ^{2}}{x ^{3} + 9 x} + \frac{2}{x ^{3} + 9 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{9 x ^{2}}{x ^{3} + 9 x} d x + \int \frac{2}{x ^{3} + 9 x} d x

\int \frac{9 x ^{2}}{x ^{3} + 9 x} d x = \frac{9}{2} ln x^{2} + 9

\int \frac{2}{x ^{3} + 9 x} d x = 2 (\frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln x^{2} + 9)

= - \frac{9}{2} ln x^{2} + 9 + 2 (\frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln x^{2} + 9)

簡素化

- \frac{9}{2} ln x^{2} + 9 + 2 (\frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln x^{2} + 9) : - \frac{83}{18} ln x^{2} + 9 + \frac{2}{9} ln ∣ x ∣

= - \frac{83}{18} ln x^{2} + 9 + \frac{2}{9} ln ∣ x ∣

定数を解答に追加する

= - \frac{83}{18} ln x^{2} + 9 + \frac{2}{9} ln ∣ x ∣ + C