What is the integral of (5x-9)/(3x^2+5x+2) ?

The integral of (5x-9)/(3x^2+5x+2) is 5/6 ln|36x^2+60x+24|+79/6 ln|6x+6|-79/6 ln|6x+4|+C

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integral of (5x-9)/(3x^2+5x+2)

\int \frac{5 x - 9}{3 x ^{2} + 5 x + 2} d x

\frac{5}{6} ln 36 x^{2} + 60 x + 24 + \frac{79}{6} ln ∣ 6 x + 6 ∣ - \frac{79}{6} ln ∣ 6 x + 4 ∣ + C

Solution steps

\int \frac{5 x - 9}{3 x ^{2} + 5 x + 2} d x

Expand

\frac{5 x - 9}{3 x ^{2} + 5 x + 2} : \frac{5 x}{3 x ^{2} + 5 x + 2} - \frac{9}{3 x ^{2} + 5 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{5 x}{3 x ^{2} + 5 x + 2} d x - \int \frac{9}{3 x ^{2} + 5 x + 2} d x

\int \frac{5 x}{3 x ^{2} + 5 x + 2} d x = 10 (\frac{1}{12} ln 36 x^{2} + 60 x + 24 + \frac{5}{12} (ln ∣ 6 x + 6 ∣ - ln ∣ 6 x + 4 ∣))

\int \frac{9}{3 x ^{2} + 5 x + 2} d x = - 18 (\frac{1}{2} ln ∣ 6 x + 6 ∣ - \frac{1}{2} ln ∣ 6 x + 4 ∣)

= 10 (\frac{1}{12} ln 36 x^{2} + 60 x + 24 + \frac{5}{12} (ln ∣ 6 x + 6 ∣ - ln ∣ 6 x + 4 ∣)) - (- 18 (\frac{1}{2} ln ∣ 6 x + 6 ∣ - \frac{1}{2} ln ∣ 6 x + 4 ∣))

Simplify

10 (\frac{1}{12} ln 36 x^{2} + 60 x + 24 + \frac{5}{12} (ln ∣ 6 x + 6 ∣ - ln ∣ 6 x + 4 ∣)) - (- 18 (\frac{1}{2} ln ∣ 6 x + 6 ∣ - \frac{1}{2} ln ∣ 6 x + 4 ∣)) : \frac{5}{6} ln 36 x^{2} + 60 x + 24 + \frac{79}{6} ln ∣ 6 x + 6 ∣ - \frac{79}{6} ln ∣ 6 x + 4 ∣

= \frac{5}{6} ln 36 x^{2} + 60 x + 24 + \frac{79}{6} ln ∣ 6 x + 6 ∣ - \frac{79}{6} ln ∣ 6 x + 4 ∣

Add a constant to the solution

= \frac{5}{6} ln 36 x^{2} + 60 x + 24 + \frac{79}{6} ln ∣ 6 x + 6 ∣ - \frac{79}{6} ln ∣ 6 x + 4 ∣ + C

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What is the integral of (5x-9)/(3x^2+5x+2) ?
The integral of (5x-9)/(3x^2+5x+2) is 5/6 ln|36x^2+60x+24|+79/6 ln|6x+6|-79/6 ln|6x+4|+C