integral of ((9+sqrt(x)+x))/x

解答

\int \frac{( 9 + x + x )}{x} d x

9 ln ∣ x ∣ + 2 x + x + C

求解步骤

\int \frac{9 + x + x}{x} d x

乘开

\frac{9 + x + x}{x} : \frac{9}{x} + \frac{1}{x} + 1

= \int \frac{9}{x} + \frac{1}{x} + 1 d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{9}{x} d x + \int \frac{1}{x} d x + \int 1 d x

\int \frac{9}{x} d x = 9 ln ∣ x ∣

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

\int 1 d x = x

= 9 ln ∣ x ∣ + 2 x + x

解答补常数

= 9 ln ∣ x ∣ + 2 x + x + C