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

解

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

x + ln ∣ x ∣ + \frac{6}{x} + C

解答ステップ

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

拡張

\frac{( x - 2 ) ( x + 3 )}{x ^{2}} : 1 + \frac{1}{x} - \frac{6}{x ^{2}}

= \int 1 + \frac{1}{x} - \frac{6}{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 1 d x + \int \frac{1}{x} d x - \int \frac{6}{x ^{2}} d x

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

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

= x + ln ∣ x ∣ - (- \frac{6}{x})

簡素化

= x + ln ∣ x ∣ + \frac{6}{x}

定数を解答に追加する

= x + ln ∣ x ∣ + \frac{6}{x} + C