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

The integral of (5x^3+3x-2)/((x-2)(x+3)) is

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

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

\frac{5 x ^{2}}{2} - 5 x + \frac{19}{2} ln \frac{2}{5} x + \frac{6}{5} - \frac{19}{2} ln \frac{2}{5} x - \frac{4}{5} + 19 ln 4 x^{2} + 4 x - 24 - \frac{2}{5} ln ∣ x - 2 ∣ + \frac{2}{5} ln ∣ x + 3 ∣ + \frac{3}{10} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + C

Solution steps

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

Expand

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

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

\int \frac{5 x ^{3}}{( x - 2 ) ( x + 3 )} d x = \frac{5 x ^{2}}{2} - 5 x + \frac{35}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + 6 ln \frac{2}{5} x + \frac{6}{5} - 6 ln \frac{2}{5} x - \frac{4}{5}

\int \frac{3 x}{( x - 2 ) ( x + 3 )} d x = 6 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x))

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

= \frac{5 x ^{2}}{2} - 5 x + \frac{35}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + 6 ln \frac{2}{5} x + \frac{6}{5} - 6 ln \frac{2}{5} x - \frac{4}{5} + 6 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) - 2 (\frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{5} ln ∣ x + 3 ∣)

Simplify

\frac{5 x ^{2}}{2} - 5 x + \frac{35}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + 6 ln \frac{2}{5} x + \frac{6}{5} - 6 ln \frac{2}{5} x - \frac{4}{5} + 6 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) - 2 (\frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{5} ln ∣ x + 3 ∣) : \frac{5 x ^{2}}{2} - 5 x + \frac{19}{2} ln \frac{2}{5} x + \frac{6}{5} - \frac{19}{2} ln \frac{2}{5} x - \frac{4}{5} + 19 ln 4 x^{2} + 4 x - 24 - \frac{2}{5} ln ∣ x - 2 ∣ + \frac{2}{5} ln ∣ x + 3 ∣ + \frac{3}{10} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

= \frac{5 x ^{2}}{2} - 5 x + \frac{19}{2} ln \frac{2}{5} x + \frac{6}{5} - \frac{19}{2} ln \frac{2}{5} x - \frac{4}{5} + 19 ln 4 x^{2} + 4 x - 24 - \frac{2}{5} ln ∣ x - 2 ∣ + \frac{2}{5} ln ∣ x + 3 ∣ + \frac{3}{10} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

Add a constant to the solution

= \frac{5 x ^{2}}{2} - 5 x + \frac{19}{2} ln \frac{2}{5} x + \frac{6}{5} - \frac{19}{2} ln \frac{2}{5} x - \frac{4}{5} + 19 ln 4 x^{2} + 4 x - 24 - \frac{2}{5} ln ∣ x - 2 ∣ + \frac{2}{5} ln ∣ x + 3 ∣ + \frac{3}{10} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + C

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