integral of sqrt(x)(sqrt(a)-sqrt(x))^2

解

\int x (a - x)^{2} d x

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

解答ステップ

\int x (a - x)^{2} d x

拡張

x (a - x)^{2} : a x - 2 a x + x^{\frac{3}{2}}

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

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

\int 2 a x d x = a x^{2}

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

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

定数を解答に追加する

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