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

解

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

- 25 arctan (\frac{x}{5}) + 45 \frac{x}{5} + C

解答ステップ

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

拡張

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

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

\int \frac{2 x}{x ( x + 5 )} d x = 45 (- arctan (\frac{x}{5}) + \frac{x}{5})

\int \frac{5}{x ( x + 5 )} d x = 25 arctan (\frac{x}{5})

= 45 (- arctan (\frac{x}{5}) + \frac{x}{5}) + 25 arctan (\frac{x}{5})

簡素化

45 (- arctan (\frac{x}{5}) + \frac{x}{5}) + 25 arctan (\frac{x}{5}) : - 25 arctan (\frac{x}{5}) + 45 \frac{x}{5}

= - 25 arctan (\frac{x}{5}) + 45 \frac{x}{5}

定数を解答に追加する

= - 25 arctan (\frac{x}{5}) + 45 \frac{x}{5} + C