What is the integral of (3x^2+7x-5)/(3x+2) ?

The integral of (3x^2+7x-5)/(3x+2) is (x^2)/2+5/3 x-25/9 ln|3x+2|+14/9+C

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integral of (3x^2+7x-5)/(3x+2)

\int \frac{3 x ^{2} + 7 x - 5}{3 x + 2} d x

\frac{x ^{2}}{2} + \frac{5}{3} x - \frac{25}{9} ln ∣ 3 x + 2 ∣ + \frac{14}{9} + C

Solution steps

\int \frac{3 x ^{2} + 7 x - 5}{3 x + 2} d x

Expand

\frac{3 x ^{2} + 7 x - 5}{3 x + 2} : \frac{3 x ^{2}}{3 x + 2} + \frac{7 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 ^{2}}{3 x + 2} d x + \int \frac{7 x}{3 x + 2} d x - \int \frac{5}{3 x + 2} d x

\int \frac{3 x ^{2}}{3 x + 2} d x = 3 (\frac{x ^{2}}{6} - \frac{2}{9} x + \frac{4}{27} ln ∣ 3 x + 2 ∣)

\int \frac{7 x}{3 x + 2} d x = \frac{7}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣)

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

= 3 (\frac{x ^{2}}{6} - \frac{2}{9} x + \frac{4}{27} ln ∣ 3 x + 2 ∣) + \frac{7}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣) - \frac{5}{3} ln ∣ 3 x + 2 ∣

Simplify

3 (\frac{x ^{2}}{6} - \frac{2}{9} x + \frac{4}{27} ln ∣ 3 x + 2 ∣) + \frac{7}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣) - \frac{5}{3} ln ∣ 3 x + 2 ∣ : \frac{x ^{2}}{2} + \frac{5}{3} x - \frac{25}{9} ln ∣ 3 x + 2 ∣ + \frac{14}{9}

= \frac{x ^{2}}{2} + \frac{5}{3} x - \frac{25}{9} ln ∣ 3 x + 2 ∣ + \frac{14}{9}

Add a constant to the solution

= \frac{x ^{2}}{2} + \frac{5}{3} x - \frac{25}{9} ln ∣ 3 x + 2 ∣ + \frac{14}{9} + C

What is the integral of (3x^2+7x-5)/(3x+2) ?
The integral of (3x^2+7x-5)/(3x+2) is (x^2)/2+5/3 x-25/9 ln|3x+2|+14/9+C