integral of (a-x)sqrt(a+x)

解

\int (a - x) a + x d x

\frac{4}{3} a (x + a)^{\frac{3}{2}} - \frac{2}{5} (x + a)^{\frac{5}{2}} + C

解答ステップ

\int (a - x) a + x d x

拡張

(a - x) a + x : a a + x - x a + x

= \int a a + x - x a + 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 a a + x d x - \int x a + x d x

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

\int x a + x d x = \frac{2}{5} (a + x)^{\frac{5}{2}} - a \frac{2}{3} (a + x)^{\frac{3}{2}}

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

簡素化

a \frac{2}{3} (a + x)^{\frac{3}{2}} - (\frac{2}{5} (a + x)^{\frac{5}{2}} - a \frac{2}{3} (a + x)^{\frac{3}{2}}) : \frac{4}{3} a (x + a)^{\frac{3}{2}} - \frac{2}{5} (x + a)^{\frac{5}{2}}

= \frac{4}{3} a (x + a)^{\frac{3}{2}} - \frac{2}{5} (x + a)^{\frac{5}{2}}

定数を解答に追加する

= \frac{4}{3} a (x + a)^{\frac{3}{2}} - \frac{2}{5} (x + a)^{\frac{5}{2}} + C