integral of (5+x+x^2)/(sqrt(x))

解

\int \frac{5 + x + x ^{2}}{x} d x

10 x + \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

解答ステップ

\int \frac{5 + x + x ^{2}}{x} d x

拡張

\frac{5 + x + x ^{2}}{x} : \frac{5}{x} + x + x^{\frac{3}{2}}

= \int \frac{5}{x} + x + x^{\frac{3}{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

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

\int \frac{5}{x} d x = 10 x

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

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

= 10 x + \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}}

定数を解答に追加する

= 10 x + \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C