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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

Add a constant to the solution

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

\int 3 x^{3} + 4 x^{2} - 2 x^{- 1} + 13 d x

\int - y^{3} d y

\frac{d}{d x} (2 e^{x} - 5 cos (x) + 3)

\frac{\partial}{\partial x} (- 6 y^{3} + 10 x^{2} - 4 x y)

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