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

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

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

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

2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \int \frac{2}{x + 1} + \frac{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{2}{x + 1} d x + \int \frac{3}{x - 3} d x

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

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

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

\frac{\partial}{\partial x} (\frac{x + y}{x - y})

\frac{d}{d x} (sin (cos (4 x)))

\frac{1}{5 x ^{\frac{4}{5}}}

f (x) = \frac{x ^{3} + 1}{x ^{3} - 1}

x^{3} y^{^{'''}} + x y^{^{'}} - y = 0

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