integral of \sqrt[7]{x^2}+xsqrt(x)

解

\int 7 x^{2} + x x d x

\frac{7}{9} x^{\frac{9}{7}} + \frac{2}{5} x^{\frac{5}{2}} + C

解答ステップ

\int 7 x^{2} + x x d x

7 x^{2} = x^{\frac{2}{7}}, （ x \geq 0

と仮定

）

= \int x^{\frac{2}{7}} + x 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 x^{\frac{2}{7}} d x + \int x x d x

\int x^{\frac{2}{7}} d x = \frac{7}{9} x^{\frac{9}{7}}

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

= \frac{7}{9} x^{\frac{9}{7}} + \frac{2}{5} x^{\frac{5}{2}}

定数を解答に追加する

= \frac{7}{9} x^{\frac{9}{7}} + \frac{2}{5} x^{\frac{5}{2}} + C