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

The integral of (3x+2)/(2x-1) is 3/2 x-3/4+7/4 ln|2x-1|+C

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

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

\frac{3}{2} x - \frac{3}{4} + \frac{7}{4} ln ∣ 2 x - 1 ∣ + C

Solution steps

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

Expand

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

\int \frac{3 x}{2 x - 1} d x = \frac{3}{4} (2 x - 1 + ln ∣ 2 x - 1 ∣)

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

= \frac{3}{4} (2 x - 1 + ln ∣ 2 x - 1 ∣) + ln ∣ 2 x - 1 ∣

Simplify

\frac{3}{4} (2 x - 1 + ln ∣ 2 x - 1 ∣) + ln ∣ 2 x - 1 ∣ : \frac{3}{2} x - \frac{3}{4} + \frac{7}{4} ln ∣ 2 x - 1 ∣

= \frac{3}{2} x - \frac{3}{4} + \frac{7}{4} ln ∣ 2 x - 1 ∣

Add a constant to the solution

= \frac{3}{2} x - \frac{3}{4} + \frac{7}{4} ln ∣ 2 x - 1 ∣ + C

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What is the integral of (3x+2)/(2x-1) ?
The integral of (3x+2)/(2x-1) is 3/2 x-3/4+7/4 ln|2x-1|+C