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

The integral of (2x^4+3)/(x^3-3x+2) is x^2+35/9 ln|x+2|+19/9 ln|x-1|-3/(x-1)+4/(3(x-1))+C

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

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

x^{2} + \frac{35}{9} ln ∣ x + 2 ∣ + \frac{19}{9} ln ∣ x - 1 ∣ - \frac{3}{x - 1} + \frac{4}{3 ( x - 1 )} + C

Solution steps

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

Long division

\frac{2 x ^{4} + 3}{x ^{3} - 3 x + 2} : 2 x + \frac{6 x ^{2} - 4 x + 3}{x ^{3} - 3 x + 2}

= \int 2 x + \frac{6 x ^{2} - 4 x + 3}{x ^{3} - 3 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 2 x d x + \int \frac{6 x ^{2} - 4 x + 3}{x ^{3} - 3 x + 2} d x

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

\int \frac{6 x ^{2} - 4 x + 3}{x ^{3} - 3 x + 2} d x = \frac{35}{9} ln ∣ x + 2 ∣ + \frac{19}{9} ln ∣ x - 1 ∣ - \frac{3}{x - 1} + \frac{4}{3 ( x - 1 )}

= x^{2} + \frac{35}{9} ln ∣ x + 2 ∣ + \frac{19}{9} ln ∣ x - 1 ∣ - \frac{3}{x - 1} + \frac{4}{3 ( x - 1 )}

Add a constant to the solution

= x^{2} + \frac{35}{9} ln ∣ x + 2 ∣ + \frac{19}{9} ln ∣ x - 1 ∣ - \frac{3}{x - 1} + \frac{4}{3 ( x - 1 )} + C

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What is the integral of (2x^4+3)/(x^3-3x+2) ?
The integral of (2x^4+3)/(x^3-3x+2) is x^2+35/9 ln|x+2|+19/9 ln|x-1|-3/(x-1)+4/(3(x-1))+C