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人気のある微分積分 >

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

解

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

解

\frac{2}{7} (x - 5)^{\frac{7}{2}} + 4 (x - 5)^{\frac{5}{2}} + \frac{50}{3} (x - 5)^{\frac{3}{2}} + C

解答ステップ

\int x^{2} x - 5 d x

u置換積分法を適用する

= \int (u + 5)^{2} u d u

拡張

(u + 5)^{2} u : u^{\frac{5}{2}} + 10 u^{\frac{3}{2}} + 25 u

= \int u^{\frac{5}{2}} + 10 u^{\frac{3}{2}} + 25 u d u

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 u^{\frac{5}{2}} d u + \int 10 u^{\frac{3}{2}} d u + \int 25 u d u

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

\int 10 u^{\frac{3}{2}} d u = 4 u^{\frac{5}{2}}

\int 25 u d u = \frac{50}{3} u^{\frac{3}{2}}

= \frac{2}{7} u^{\frac{7}{2}} + 4 u^{\frac{5}{2}} + \frac{50}{3} u^{\frac{3}{2}}

代用を戻す

u = x - 5

= \frac{2}{7} (x - 5)^{\frac{7}{2}} + 4 (x - 5)^{\frac{5}{2}} + \frac{50}{3} (x - 5)^{\frac{3}{2}}

定数を解答に追加する

= \frac{2}{7} (x - 5)^{\frac{7}{2}} + 4 (x - 5)^{\frac{5}{2}} + \frac{50}{3} (x - 5)^{\frac{3}{2}} + C

グラフ

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