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

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

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

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

\frac{x ^{2}}{2} - x + 14 (\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)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + C

Solution steps

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

Long division

\frac{x ^{3}}{( x - 2 ) ( x + 3 )} : x - 1 + \frac{7 x - 6}{x ^{2} + x - 6}

= \int x - 1 + \frac{7 x - 6}{x ^{2} + x - 6} 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 x d x - \int 1 d x + \int \frac{7 x - 6}{x ^{2} + x - 6} d x

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

\int 1 d x = x

\int \frac{7 x - 6}{x ^{2} + x - 6} d x = 14 (\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)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

= \frac{x ^{2}}{2} - x + 14 (\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)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

Add a constant to the solution

= \frac{x ^{2}}{2} - x + 14 (\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)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + C

y = \frac{x ^{2}}{x + 3}, (2, \frac{4}{5})

f (x) = 4 x^{2} + 1

y^{^{'}} + \frac{1}{2} y = 8, y (0) = 2

\frac{\partial}{\partial x} (\frac{( x v ^{2} )}{( x ^{2} + v ^{2} )})

(\frac{x + 2}{x}) (7 - 2 x^{3})

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