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

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

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

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

x - \frac{1}{2} + ln ∣ 2 x - 1 ∣ + C

Solution steps

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

Expand

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

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

\int \frac{2 x}{2 x - 1} d x = x - \frac{1}{2} + \frac{1}{2} ln ∣ 2 x - 1 ∣

\int \frac{1}{2 x - 1} d x = \frac{1}{2} ln ∣ 2 x - 1 ∣

= x - \frac{1}{2} + \frac{1}{2} ln ∣ 2 x - 1 ∣ + \frac{1}{2} ln ∣ 2 x - 1 ∣

Add similar elements:

\frac{1}{2} ln ∣ 2 x - 1 ∣ + \frac{1}{2} ln ∣ 2 x - 1 ∣ = ln ∣ 2 x - 1 ∣

= x - \frac{1}{2} + ln ∣ 2 x - 1 ∣

Add a constant to the solution

= x - \frac{1}{2} + ln ∣ 2 x - 1 ∣ + C

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