What is the integral of (x^5)/(x^3-1) ?

The integral of (x^5)/(x^3-1) is (x^3)/3+1/3 ln|x^3-1|+C

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integral of (x^5)/(x^3-1)

\int \frac{x ^{5}}{x ^{3} - 1} d x

\frac{x ^{3}}{3} + \frac{1}{3} ln x^{3} - 1 + C

Solution steps

\int \frac{x ^{5}}{x ^{3} - 1} d x

Long division

\frac{x ^{5}}{x ^{3} - 1} : x^{2} + \frac{x ^{2}}{x ^{3} - 1}

= \int x^{2} + \frac{x ^{2}}{x ^{3} - 1} 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^{2} d x + \int \frac{x ^{2}}{x ^{3} - 1} d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int \frac{x ^{2}}{x ^{3} - 1} d x = \frac{1}{3} ln x^{3} - 1

= \frac{x ^{3}}{3} + \frac{1}{3} ln x^{3} - 1

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{1}{3} ln x^{3} - 1 + C

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What is the integral of (x^5)/(x^3-1) ?
The integral of (x^5)/(x^3-1) is (x^3)/3+1/3 ln|x^3-1|+C