integral of (9x^3-x^2+45x+5)/(x^3+5x)

解

\int \frac{9 x ^{3} - x ^{2} + 45 x + 5}{x ^{3} + 5 x} d x

9 x - ln x^{2} + 5 + ln ∣ x ∣ + C

解答ステップ

\int \frac{9 x ^{3} - x ^{2} + 45 x + 5}{x ^{3} + 5 x} d x

拡張

\frac{9 x ^{3} - x ^{2} + 45 x + 5}{x ^{3} + 5 x} : \frac{9 x ^{3}}{x ^{3} + 5 x} - \frac{x ^{2}}{x ^{3} + 5 x} + \frac{45 x}{x ^{3} + 5 x} + \frac{5}{x ^{3} + 5 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 ^{3}}{x ^{3} + 5 x} d x - \int \frac{x ^{2}}{x ^{3} + 5 x} d x + \int \frac{45 x}{x ^{3} + 5 x} d x + \int \frac{5}{x ^{3} + 5 x} d x

\int \frac{9 x ^{3}}{x ^{3} + 5 x} d x = - 95 arctan (\frac{x}{5}) + 9 x

\int \frac{x ^{2}}{x ^{3} + 5 x} d x = \frac{1}{2} ln x^{2} + 5

\int \frac{45 x}{x ^{3} + 5 x} d x = 95 arctan (\frac{x}{5})

\int \frac{5}{x ^{3} + 5 x} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 5

= - 95 arctan (\frac{x}{5}) + 9 x - \frac{1}{2} ln x^{2} + 5 + 95 arctan (\frac{x}{5}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 5

簡素化

- 95 arctan (\frac{x}{5}) + 9 x - \frac{1}{2} ln x^{2} + 5 + 95 arctan (\frac{x}{5}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 5 : 9 x - ln x^{2} + 5 + ln ∣ x ∣

= 9 x - ln x^{2} + 5 + ln ∣ x ∣

定数を解答に追加する

= 9 x - ln x^{2} + 5 + ln ∣ x ∣ + C