What is the integral of ((2+sqrt(x)+x))/x ?

The integral of ((2+sqrt(x)+x))/x is 2ln|x|+2sqrt(x)+x+C

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

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

2 ln ∣ x ∣ + 2 x + x + C

Solution steps

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

Expand

\frac{2 + x + x}{x} : \frac{2}{x} + \frac{1}{x} + 1

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

Apply the Sum Rule:

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

\int 1 d x = x

= 2 ln ∣ x ∣ + 2 x + x

Add a constant to the solution

= 2 ln ∣ x ∣ + 2 x + x + C

What is the integral of ((2+sqrt(x)+x))/x ?
The integral of ((2+sqrt(x)+x))/x is 2ln|x|+2sqrt(x)+x+C