What is the integral of ((2x+1))/x ?

The integral of ((2x+1))/x is 2x+ln|x|+C

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integral of ((2x+1))/x

\int \frac{( 2 x + 1 )}{x} d x

2 x + ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

\int 2 d x = 2 x

\int \frac{1}{x} d x = ln ∣ x ∣

= 2 x + ln ∣ x ∣

Add a constant to the solution

= 2 x + ln ∣ x ∣ + C