integral of (7x^3-5x^2+7x-8)/(x^2)

解

\int \frac{7 x ^{3} - 5 x ^{2} + 7 x - 8}{x ^{2}} d x

\frac{7 x ^{2}}{2} - 5 x + 7 ln ∣ x ∣ + \frac{8}{x} + C

解答ステップ

\int \frac{7 x ^{3} - 5 x ^{2} + 7 x - 8}{x ^{2}} d x

拡張

\frac{7 x ^{3} - 5 x ^{2} + 7 x - 8}{x ^{2}} : 7 x - 5 + \frac{7}{x} - \frac{8}{x ^{2}}

= \int 7 x - 5 + \frac{7}{x} - \frac{8}{x ^{2}} 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 7 x d x - \int 5 d x + \int \frac{7}{x} d x - \int \frac{8}{x ^{2}} d x

\int 7 x d x = \frac{7 x ^{2}}{2}

\int 5 d x = 5 x

\int \frac{7}{x} d x = 7 ln ∣ x ∣

\int \frac{8}{x ^{2}} d x = - \frac{8}{x}

= \frac{7 x ^{2}}{2} - 5 x + 7 ln ∣ x ∣ - (- \frac{8}{x})

簡素化

= \frac{7 x ^{2}}{2} - 5 x + 7 ln ∣ x ∣ + \frac{8}{x}

定数を解答に追加する

= \frac{7 x ^{2}}{2} - 5 x + 7 ln ∣ x ∣ + \frac{8}{x} + C