integral of ((5x-3)^3)/(x^3)

解

\int \frac{( 5 x - 3 ) ^{3}}{x ^{3}} d x

125 x - 225 ln ∣ x ∣ - \frac{135}{x} + \frac{27}{2 x ^{2}} + C

解答ステップ

\int \frac{( 5 x - 3 ) ^{3}}{x ^{3}} d x

長除法

\frac{( 5 x - 3 ) ^{3}}{x ^{3}} : 125 + \frac{- 225 x ^{2} + 135 x - 27}{x ^{3}}

= \int 125 + \frac{- 225 x ^{2} + 135 x - 27}{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 125 d x + \int \frac{- 225 x ^{2} + 135 x - 27}{x ^{3}} d x

\int 125 d x = 125 x

\int \frac{- 225 x ^{2} + 135 x - 27}{x ^{3}} d x = - 225 ln ∣ x ∣ - \frac{135}{x} + \frac{27}{2 x ^{2}}

= 125 x - 225 ln ∣ x ∣ - \frac{135}{x} + \frac{27}{2 x ^{2}}

定数を解答に追加する

= 125 x - 225 ln ∣ x ∣ - \frac{135}{x} + \frac{27}{2 x ^{2}} + C