What is the integral of (4x+3)/(2x+1) ?

The integral of (4x+3)/(2x+1) is 2x+1/2 ln|2x+1|+C

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integral of (4x+3)/(2x+1)

\int \frac{4 x + 3}{2 x + 1} d x

2 x + \frac{1}{2} ln ∣ 2 x + 1 ∣ + C

Solution steps

\int \frac{4 x + 3}{2 x + 1} d x

Long division

\frac{4 x + 3}{2 x + 1} : 2 + \frac{1}{2 x + 1}

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

\int 2 d x = 2 x

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

= 2 x + \frac{1}{2} ln ∣ 2 x + 1 ∣

Add a constant to the solution

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

What is the integral of (4x+3)/(2x+1) ?
The integral of (4x+3)/(2x+1) is 2x+1/2 ln|2x+1|+C