integral of (3x^5-6x^2+2x-5)/(x^3)

解

\int \frac{3 x ^{5} - 6 x ^{2} + 2 x - 5}{x ^{3}} d x

x^{3} - 6 ln ∣ x ∣ - \frac{2}{x} + \frac{5}{2 x ^{2}} + C

解答ステップ

\int \frac{3 x ^{5} - 6 x ^{2} + 2 x - 5}{x ^{3}} d x

拡張

\frac{3 x ^{5} - 6 x ^{2} + 2 x - 5}{x ^{3}} : 3 x^{2} - \frac{6}{x} + \frac{2}{x ^{2}} - \frac{5}{x ^{3}}

= \int 3 x^{2} - \frac{6}{x} + \frac{2}{x ^{2}} - \frac{5}{x ^{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 3 x^{2} d x - \int \frac{6}{x} d x + \int \frac{2}{x ^{2}} d x - \int \frac{5}{x ^{3}} d x

\int 3 x^{2} d x = x^{3}

\int \frac{6}{x} d x = 6 ln ∣ x ∣

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

\int \frac{5}{x ^{3}} d x = - \frac{5}{2 x ^{2}}

= x^{3} - 6 ln ∣ x ∣ - \frac{2}{x} - (- \frac{5}{2 x ^{2}})

簡素化

= x^{3} - 6 ln ∣ x ∣ - \frac{2}{x} + \frac{5}{2 x ^{2}}

定数を解答に追加する

= x^{3} - 6 ln ∣ x ∣ - \frac{2}{x} + \frac{5}{2 x ^{2}} + C