What is the integral of (6x+2)/((x+1)(2x+1)) ?

The integral of (6x+2)/((x+1)(2x+1)) is 6ln|2x+2|-ln|2x+1|-2ln|x+1|+C

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integral of (6x+2)/((x+1)(2x+1))

\int \frac{6 x + 2}{( x + 1 ) ( 2 x + 1 )} d x

6 ln ∣ 2 x + 2 ∣ - ln ∣ 2 x + 1 ∣ - 2 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{6 x + 2}{( x + 1 ) ( 2 x + 1 )} d x

Expand

\frac{6 x + 2}{( x + 1 ) ( 2 x + 1 )} : \frac{6 x}{( x + 1 ) ( 2 x + 1 )} + \frac{2}{( x + 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{6 x}{( x + 1 ) ( 2 x + 1 )} d x + \int \frac{2}{( x + 1 ) ( 2 x + 1 )} d x

\int \frac{6 x}{( x + 1 ) ( 2 x + 1 )} d x = 3 (2 ln ∣ 2 x + 2 ∣ - ln ∣ 2 x + 1 ∣)

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

= 3 (2 ln ∣ 2 x + 2 ∣ - ln ∣ 2 x + 1 ∣) + 2 (- ln ∣ x + 1 ∣ + ln ∣ 2 x + 1 ∣)

Simplify

3 (2 ln ∣ 2 x + 2 ∣ - ln ∣ 2 x + 1 ∣) + 2 (- ln ∣ x + 1 ∣ + ln ∣ 2 x + 1 ∣) : 6 ln ∣ 2 x + 2 ∣ - ln ∣ 2 x + 1 ∣ - 2 ln ∣ x + 1 ∣

= 6 ln ∣ 2 x + 2 ∣ - ln ∣ 2 x + 1 ∣ - 2 ln ∣ x + 1 ∣

Add a constant to the solution

= 6 ln ∣ 2 x + 2 ∣ - ln ∣ 2 x + 1 ∣ - 2 ln ∣ x + 1 ∣ + C

What is the integral of (6x+2)/((x+1)(2x+1)) ?
The integral of (6x+2)/((x+1)(2x+1)) is 6ln|2x+2|-ln|2x+1|-2ln|x+1|+C