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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

Add a constant to the solution

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

\int (2 t^{2} + 1) d t

\int r \cdot e^{\frac{r}{2}} d r

\int \frac{3 x}{3 x + 5} d x

\int 2 2 x + 3 d x

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