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

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

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

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

2 x + 1 + ln ∣ 2 x + 1 ∣ + C

Solution steps

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

Expand

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

\int \frac{4 x}{2 x + 1} d x = 2 x + 1 - ln ∣ 2 x + 1 ∣

\int \frac{4}{2 x + 1} d x = 2 ln ∣ 2 x + 1 ∣

= 2 x + 1 - ln ∣ 2 x + 1 ∣ + 2 ln ∣ 2 x + 1 ∣

Simplify

= 2 x + 1 + ln ∣ 2 x + 1 ∣

Add a constant to the solution

= 2 x + 1 + ln ∣ 2 x + 1 ∣ + C

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