integral of x^2(x-5)sqrt(7-x)

解

\int x^{2} (x - 5) 7 - x d x

\frac{2}{9} (- x + 7)^{\frac{9}{2}} - \frac{32}{7} (- x + 7)^{\frac{7}{2}} + \frac{154}{5} (- x + 7)^{\frac{5}{2}} - \frac{196}{3} (- x + 7)^{\frac{3}{2}} + C

解答ステップ

\int x^{2} (x - 5) 7 - x d x

拡張

x^{2} (x - 5) 7 - x : x^{3} 7 - x - 5 x^{2} 7 - x

= \int x^{3} 7 - x - 5 x^{2} 7 - 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^{3} 7 - x d x - \int 5 x^{2} 7 - x d x

\int x^{3} 7 - x d x = \frac{2}{9} (7 - x)^{\frac{9}{2}} - 6 (7 - x)^{\frac{7}{2}} + \frac{294}{5} (7 - x)^{\frac{5}{2}} - \frac{686}{3} (7 - x)^{\frac{3}{2}}

\int 5 x^{2} 7 - x d x = - \frac{10}{7} (7 - x)^{\frac{7}{2}} + 28 (7 - x)^{\frac{5}{2}} - \frac{490}{3} (7 - x)^{\frac{3}{2}}

= \frac{2}{9} (7 - x)^{\frac{9}{2}} - 6 (7 - x)^{\frac{7}{2}} + \frac{294}{5} (7 - x)^{\frac{5}{2}} - \frac{686}{3} (7 - x)^{\frac{3}{2}} - (- \frac{10}{7} (7 - x)^{\frac{7}{2}} + 28 (7 - x)^{\frac{5}{2}} - \frac{490}{3} (7 - x)^{\frac{3}{2}})

簡素化

\frac{2}{9} (7 - x)^{\frac{9}{2}} - 6 (7 - x)^{\frac{7}{2}} + \frac{294}{5} (7 - x)^{\frac{5}{2}} - \frac{686}{3} (7 - x)^{\frac{3}{2}} - (- \frac{10}{7} (7 - x)^{\frac{7}{2}} + 28 (7 - x)^{\frac{5}{2}} - \frac{490}{3} (7 - x)^{\frac{3}{2}}) : \frac{2}{9} (- x + 7)^{\frac{9}{2}} - \frac{32}{7} (- x + 7)^{\frac{7}{2}} + \frac{154}{5} (- x + 7)^{\frac{5}{2}} - \frac{196}{3} (- x + 7)^{\frac{3}{2}}

= \frac{2}{9} (- x + 7)^{\frac{9}{2}} - \frac{32}{7} (- x + 7)^{\frac{7}{2}} + \frac{154}{5} (- x + 7)^{\frac{5}{2}} - \frac{196}{3} (- x + 7)^{\frac{3}{2}}

定数を解答に追加する

= \frac{2}{9} (- x + 7)^{\frac{9}{2}} - \frac{32}{7} (- x + 7)^{\frac{7}{2}} + \frac{154}{5} (- x + 7)^{\frac{5}{2}} - \frac{196}{3} (- x + 7)^{\frac{3}{2}} + C