integral of-(4x+10)/(f^2(x)*x^3)

解

\int - \frac{4 x + 10}{f ^{2} ( x ) \cdot x ^{3}}

- \frac{1}{f ^{2}} (- \frac{10}{3 x ^{3}} - \frac{2}{x ^{2}}) + C

解答ステップ

\int - \frac{4 x + 10}{f ^{2} x x ^{3}} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{f ^{2}} \cdot \int \frac{10 + 4 x}{x x ^{3}} d x

拡張

\frac{10 + 4 x}{x x ^{3}} : \frac{10}{x ^{4}} + \frac{4}{x ^{3}}

= - \frac{1}{f ^{2}} \cdot \int \frac{10}{x ^{4}} + \frac{4}{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

= - \frac{1}{f ^{2}} (\int \frac{10}{x ^{4}} d x + \int \frac{4}{x ^{3}} d x)

\int \frac{10}{x ^{4}} d x = - \frac{10}{3 x ^{3}}

\int \frac{4}{x ^{3}} d x = - \frac{2}{x ^{2}}

= - \frac{1}{f ^{2}} (- \frac{10}{3 x ^{3}} - \frac{2}{x ^{2}})

定数を解答に追加する

= - \frac{1}{f ^{2}} (- \frac{10}{3 x ^{3}} - \frac{2}{x ^{2}}) + C