What is the integral of (6x+4)/(3x^2+4x) ?

The integral of (6x+4)/(3x^2+4x) is 2ln|3x+4|-ln| 3/2 x+2|+ln(3/2 |x|)+C

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integral of (6x+4)/(3x^2+4x)

\int \frac{6 x + 4}{3 x ^{2} + 4 x} d x

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

Solution steps

\int \frac{6 x + 4}{3 x ^{2} + 4 x} d x

Expand

\frac{6 x + 4}{3 x ^{2} + 4 x} : \frac{6 x}{3 x ^{2} + 4 x} + \frac{4}{3 x ^{2} + 4 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 \frac{6 x}{3 x ^{2} + 4 x} d x + \int \frac{4}{3 x ^{2} + 4 x} d x

\int \frac{6 x}{3 x ^{2} + 4 x} d x = 2 ln ∣ 3 x + 4 ∣

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

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

Add a constant to the solution

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

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What is the integral of (6x+4)/(3x^2+4x) ?
The integral of (6x+4)/(3x^2+4x) is 2ln|3x+4|-ln| 3/2 x+2|+ln(3/2 |x|)+C