integral of ((6+sqrt(x)+x)/x)

解

\int (\frac{6 + x + x}{x}) d x

6 ln ∣ x ∣ + 2 x + x + C

解答ステップ

\int \frac{6 + x + x}{x} d x

拡張

\frac{6 + x + x}{x} : \frac{6}{x} + \frac{1}{x} + 1

= \int \frac{6}{x} + \frac{1}{x} + 1 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{6}{x} d x + \int \frac{1}{x} d x + \int 1 d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{1}{x} d x = 2 x

\int 1 d x = x

= 6 ln ∣ x ∣ + 2 x + x

定数を解答に追加する

= 6 ln ∣ x ∣ + 2 x + x + C