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Popular Calculus >

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{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) + 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{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

= \frac{x ^{2}}{2} + x + 14 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

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{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) + C

Graph

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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