integral of \sqrt[3]{x^2}-sqrt(x^3)

解

\int 3 x^{2} - x^{3} d x

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

解答ステップ

\int 3 x^{2} - 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 3 x^{2} d x - \int x^{3} d x

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

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

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

定数を解答に追加する

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