integral of (sqrt(x)+3)^3

解

\int (x + 3)^{3} d x

\frac{2}{5} x^{\frac{5}{2}} + \frac{9 x ^{2}}{2} + 18 x^{\frac{3}{2}} + 27 x + C

解答ステップ

\int (x + 3)^{3} d x

拡張

(x + 3)^{3} : x^{\frac{3}{2}} + 9 x + 27 x + 27

= \int x^{\frac{3}{2}} + 9 x + 27 x + 27 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 9 x d x + \int 27 x d x + \int 27 d x

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

\int 9 x d x = \frac{9 x ^{2}}{2}

\int 27 x d x = 18 x^{\frac{3}{2}}

\int 27 d x = 27 x

= \frac{2}{5} x^{\frac{5}{2}} + \frac{9 x ^{2}}{2} + 18 x^{\frac{3}{2}} + 27 x

定数を解答に追加する

= \frac{2}{5} x^{\frac{5}{2}} + \frac{9 x ^{2}}{2} + 18 x^{\frac{3}{2}} + 27 x + C