integral of (sqrt(x)+7)(x-sqrt(x)+1)

解

\int (x + 7) (x - x + 1) d x

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

解答ステップ

\int (x + 7) (x - x + 1) d x

拡張

(x + 7) (x - x + 1) : x^{\frac{3}{2}} + 6 x - 6 x + 7

= \int x^{\frac{3}{2}} + 6 x - 6 x + 7 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^{\frac{3}{2}} d x + \int 6 x d x - \int 6 x d x + \int 7 d x

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

\int 6 x d x = 3 x^{2}

\int 6 x d x = 4 x^{\frac{3}{2}}

\int 7 d x = 7 x

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

定数を解答に追加する

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