What is the integral of 3x+(3x-2)^{-1} ?

The integral of 3x+(3x-2)^{-1} is (3x^2)/2+1/3 ln|3x-2|+C

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integral of 3x+(3x-2)^{-1}

\int 3 x + (3 x - 2)^{- 1} d x

\frac{3 x ^{2}}{2} + \frac{1}{3} ln ∣ 3 x - 2 ∣ + C

Solution steps

\int 3 x + (3 x - 2)^{- 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 3 x d x + \int (3 x - 2)^{- 1} d x

\int 3 x d x = \frac{3 x ^{2}}{2}

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

= \frac{3 x ^{2}}{2} + \frac{1}{3} ln ∣ 3 x - 2 ∣

Add a constant to the solution

= \frac{3 x ^{2}}{2} + \frac{1}{3} ln ∣ 3 x - 2 ∣ + C

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