integral of 2sqrt(x)(10x+3)

解

\int 2 x (10 x + 3) d x

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

解答ステップ

\int 2 x (10 x + 3) 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 (10 x + 3) d x

拡張

x (10 x + 3) : 10 x^{\frac{3}{2}} + 3 x

= 2 \cdot \int 10 x^{\frac{3}{2}} + 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

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

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

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

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

定数を解答に追加する

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