integral of 2sqrt(x)(1+x^2)

解

\int 2 x (1 + x^{2}) d x

2 (\frac{2}{3} x^{\frac{3}{2}} + \frac{2}{7} x^{\frac{7}{2}}) + C

解答ステップ

\int 2 x (1 + x^{2}) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x (1 + x^{2}) d x

拡張

x (1 + x^{2}) : x + x^{\frac{5}{2}}

= 2 \cdot \int x + x^{\frac{5}{2}} 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

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

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

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

= 2 (\frac{2}{3} x^{\frac{3}{2}} + \frac{2}{7} x^{\frac{7}{2}})

定数を解答に追加する

= 2 (\frac{2}{3} x^{\frac{3}{2}} + \frac{2}{7} x^{\frac{7}{2}}) + C