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Popular >

integral of (x^3)/(x^2+x-6)

Solution

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

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

Solution steps

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

Factor

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

= \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

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Frequently Asked Questions (FAQ)

What is the integral of (x^3)/(x^2+x-6) ?
The integral of (x^3)/(x^2+x-6) is