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

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

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

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

x^{2} + 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

\int 2 x d x = x^{2}

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

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

= x^{2} + 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣

Add a constant to the solution

= x^{2} + 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

y^{^{''}} + 4 y = - 3 x^{2} + 2 x + 3

x \to 0 lim (\frac{e ^{x} - 1}{x})

y = 3 x^{2} + 5 x - 1

\frac{\partial}{\partial z} (sin (yz))

\int 12 sin^{2} (x) cos^{3} (x) d x

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