What is the integral of (6x-5)/(3x^2+5x) ?

The integral of (6x-5)/(3x^2+5x) is 2ln|3x+5|+ln| 6/5 x+2|-ln(6/5 |x|)+C

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integral of (6x-5)/(3x^2+5x)

\int \frac{6 x - 5}{3 x ^{2} + 5 x} d x

2 ln ∣ 3 x + 5 ∣ + ln \frac{6}{5} x + 2 - ln (\frac{6}{5} ∣ x ∣) + C

Solution steps

\int \frac{6 x - 5}{3 x ^{2} + 5 x} d x

Expand

\frac{6 x - 5}{3 x ^{2} + 5 x} : \frac{6 x}{3 x ^{2} + 5 x} - \frac{5}{3 x ^{2} + 5 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} + 5 x} d x - \int \frac{5}{3 x ^{2} + 5 x} d x

\int \frac{6 x}{3 x ^{2} + 5 x} d x = 2 ln ∣ 3 x + 5 ∣

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

= 2 ln ∣ 3 x + 5 ∣ - (- ln \frac{6}{5} x + 2 + ln (\frac{6}{5} ∣ x ∣))

Simplify

= 2 ln ∣ 3 x + 5 ∣ + ln \frac{6}{5} x + 2 - ln (\frac{6}{5} ∣ x ∣)

Add a constant to the solution

= 2 ln ∣ 3 x + 5 ∣ + ln \frac{6}{5} x + 2 - ln (\frac{6}{5} ∣ x ∣) + C

y = 2 x^{3} - 9 x, y = 3 x^{2} - 10

y^{^{''}} - 2 y^{^{'}} + 5 y = e^{x} cos (2 x)

\frac{d}{d x} (cos (3 x + 1))

\frac{d}{d x} (x 3 - x)

\frac{\partial}{\partial x} (- 3 x e^{2 x y})

What is the integral of (6x-5)/(3x^2+5x) ?
The integral of (6x-5)/(3x^2+5x) is 2ln|3x+5|+ln| 6/5 x+2|-ln(6/5 |x|)+C