What is the integral of (3x+5)/(3x+2) ?

The integral of (3x+5)/(3x+2) is x+2/3+ln|3x+2|+C

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integral of (3x+5)/(3x+2)

\int \frac{3 x + 5}{3 x + 2} d x

x + \frac{2}{3} + ln ∣ 3 x + 2 ∣ + C

Solution steps

\int \frac{3 x + 5}{3 x + 2} d x

Expand

\frac{3 x + 5}{3 x + 2} : \frac{3 x}{3 x + 2} + \frac{5}{3 x + 2}

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}{3 x + 2} d x + \int \frac{5}{3 x + 2} d x

\int \frac{3 x}{3 x + 2} d x = x + \frac{2}{3} - \frac{2}{3} ln ∣ 3 x + 2 ∣

\int \frac{5}{3 x + 2} d x = \frac{5}{3} ln ∣ 3 x + 2 ∣

= x + \frac{2}{3} - \frac{2}{3} ln ∣ 3 x + 2 ∣ + \frac{5}{3} ln ∣ 3 x + 2 ∣

Simplify

x + \frac{2}{3} - \frac{2}{3} ln ∣ 3 x + 2 ∣ + \frac{5}{3} ln ∣ 3 x + 2 ∣ : x + \frac{2}{3} + ln ∣ 3 x + 2 ∣

= x + \frac{2}{3} + ln ∣ 3 x + 2 ∣

Add a constant to the solution

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

What is the integral of (3x+5)/(3x+2) ?
The integral of (3x+5)/(3x+2) is x+2/3+ln|3x+2|+C