What is the integral of (4x-5)/(2x^3-5x^2-3x) ?

The integral of (4x-5)/(2x^3-5x^2-3x) is 5/3 ln|x|-2ln|2x+1|+1/3 ln|x-3|+C

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integral of (4x-5)/(2x^3-5x^2-3x)

\int \frac{4 x - 5}{2 x ^{3} - 5 x ^{2} - 3 x} d x

\frac{5}{3} ln ∣ x ∣ - 2 ln ∣ 2 x + 1 ∣ + \frac{1}{3} ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{4 x - 5}{2 x ^{3} - 5 x ^{2} - 3 x} d x

Take the partial fraction of

\frac{4 x - 5}{2 x ^{3} - 5 x ^{2} - 3 x} : \frac{5}{3 x} - \frac{4}{2 x + 1} + \frac{1}{3 ( x - 3 )}

= \int \frac{5}{3 x} - \frac{4}{2 x + 1} + \frac{1}{3 ( x - 3 )} d 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{5}{3 x} d x - \int \frac{4}{2 x + 1} d x + \int \frac{1}{3 ( x - 3 )} d x

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

\int \frac{4}{2 x + 1} d x = 2 ln ∣ 2 x + 1 ∣

\int \frac{1}{3 ( x - 3 )} d x = \frac{1}{3} ln ∣ x - 3 ∣

= \frac{5}{3} ln ∣ x ∣ - 2 ln ∣ 2 x + 1 ∣ + \frac{1}{3} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{5}{3} ln ∣ x ∣ - 2 ln ∣ 2 x + 1 ∣ + \frac{1}{3} ln ∣ x - 3 ∣ + C

What is the integral of (4x-5)/(2x^3-5x^2-3x) ?
The integral of (4x-5)/(2x^3-5x^2-3x) is 5/3 ln|x|-2ln|2x+1|+1/3 ln|x-3|+C