integral of (x+5)^2sqrt(3-x)

解

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

- \frac{2}{7} (- x + 3)^{\frac{7}{2}} + \frac{32}{5} (- x + 3)^{\frac{5}{2}} - \frac{128}{3} (- x + 3)^{\frac{3}{2}} + C

解答ステップ

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

拡張

(x + 5)^{2} 3 - x : x^{2} 3 - x + 10 x 3 - x + 25 3 - x

= \int x^{2} 3 - x + 10 x 3 - x + 25 3 - 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^{2} 3 - x d x + \int 10 x 3 - x d x + \int 25 3 - x d x

\int x^{2} 3 - x d x = - \frac{2}{7} (3 - x)^{\frac{7}{2}} + \frac{12}{5} (3 - x)^{\frac{5}{2}} - 6 (3 - x)^{\frac{3}{2}}

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

\int 25 3 - x d x = - \frac{50}{3} (3 - x)^{\frac{3}{2}}

= - \frac{2}{7} (3 - x)^{\frac{7}{2}} + \frac{12}{5} (3 - x)^{\frac{5}{2}} - 6 (3 - x)^{\frac{3}{2}} - 10 (- \frac{2}{5} (3 - x)^{\frac{5}{2}} + 2 (3 - x)^{\frac{3}{2}}) - \frac{50}{3} (3 - x)^{\frac{3}{2}}

簡素化

- \frac{2}{7} (3 - x)^{\frac{7}{2}} + \frac{12}{5} (3 - x)^{\frac{5}{2}} - 6 (3 - x)^{\frac{3}{2}} - 10 (- \frac{2}{5} (3 - x)^{\frac{5}{2}} + 2 (3 - x)^{\frac{3}{2}}) - \frac{50}{3} (3 - x)^{\frac{3}{2}} : - \frac{2}{7} (- x + 3)^{\frac{7}{2}} + \frac{32}{5} (- x + 3)^{\frac{5}{2}} - \frac{128}{3} (- x + 3)^{\frac{3}{2}}

= - \frac{2}{7} (- x + 3)^{\frac{7}{2}} + \frac{32}{5} (- x + 3)^{\frac{5}{2}} - \frac{128}{3} (- x + 3)^{\frac{3}{2}}

定数を解答に追加する

= - \frac{2}{7} (- x + 3)^{\frac{7}{2}} + \frac{32}{5} (- x + 3)^{\frac{5}{2}} - \frac{128}{3} (- x + 3)^{\frac{3}{2}} + C