What is the integral of (2x+3)/(2x+1) ?

The integral of (2x+3)/(2x+1) is ln|2x+1|+x+C

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integral of (2x+3)/(2x+1)

\int \frac{2 x + 3}{2 x + 1} d x

ln ∣ 2 x + 1 ∣ + x + C

Solution steps

\int \frac{2 x + 3}{2 x + 1} d x

\frac{2 x + 3}{2 x + 1} = \frac{2}{2 x + 1} + 1

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

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

\int 1 d x = x

= ln ∣ 2 x + 1 ∣ + x

Add a constant to the solution

= ln ∣ 2 x + 1 ∣ + x + C

What is the integral of (2x+3)/(2x+1) ?
The integral of (2x+3)/(2x+1) is ln|2x+1|+x+C