integral of (8x-5)/(\sqrt[3]{x)}

解

\int \frac{8 x - 5}{3 x} d x

\frac{24}{5} x^{\frac{5}{3}} - \frac{15}{2} x^{\frac{2}{3}} + C

解答ステップ

\int \frac{8 x - 5}{3 x} d x

拡張

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

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

\int 8 x^{\frac{2}{3}} d x = \frac{24}{5} x^{\frac{5}{3}}

\int \frac{5}{3 x} d x = \frac{15}{2} x^{\frac{2}{3}}

= \frac{24}{5} x^{\frac{5}{3}} - \frac{15}{2} x^{\frac{2}{3}}

定数を解答に追加する

= \frac{24}{5} x^{\frac{5}{3}} - \frac{15}{2} x^{\frac{2}{3}} + C