integral of ((2x^2-x+3))/(sqrt(x))

解

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

\frac{4}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + 6 x + C

解答ステップ

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

拡張

\frac{2 x ^{2} - x + 3}{x} : 2 x^{\frac{3}{2}} - x + \frac{3}{x}

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

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

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

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

\int \frac{3}{x} d x = 6 x

= \frac{4}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + 6 x

定数を解答に追加する

= \frac{4}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + 6 x + C