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

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

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

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

x + \frac{1}{2} ln ∣ x ∣ + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Expand

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

= \frac{1}{2} \cdot \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

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

\int 2 d x = 2 x

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{1}{2} (2 x + ln ∣ x ∣)

Simplify

\frac{1}{2} (2 x + ln ∣ x ∣) : x + \frac{1}{2} ln ∣ x ∣

= x + \frac{1}{2} ln ∣ x ∣

Add a constant to the solution

= x + \frac{1}{2} ln ∣ x ∣ + C

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