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

解

\int \frac{( x + 6 ) ^{2}}{x ^{3}} d x

ln ∣ x ∣ - \frac{12}{x} - \frac{18}{x ^{2}} + C

解答ステップ

\int \frac{( x + 6 ) ^{2}}{x ^{3}} d x

拡張

\frac{( x + 6 ) ^{2}}{x ^{3}} : \frac{1}{x} + \frac{12}{x ^{2}} + \frac{36}{x ^{3}}

= \int \frac{1}{x} + \frac{12}{x ^{2}} + \frac{36}{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 \frac{1}{x} d x + \int \frac{12}{x ^{2}} d x + \int \frac{36}{x ^{3}} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{12}{x ^{2}} d x = - \frac{12}{x}

\int \frac{36}{x ^{3}} d x = - \frac{18}{x ^{2}}

= ln ∣ x ∣ - \frac{12}{x} - \frac{18}{x ^{2}}

定数を解答に追加する

= ln ∣ x ∣ - \frac{12}{x} - \frac{18}{x ^{2}} + C