What is the integral of (x^{14})/(x^5-1) ?

The integral of (x^{14})/(x^5-1) is (x^{10}}{10}+\frac{x^5)/5+1/5 ln|x^5-1|+C

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integral of (x^{14})/(x^5-1)

\int \frac{x ^{14}}{x ^{5} - 1} d x

\frac{x ^{10}}{10} + \frac{x ^{5}}{5} + \frac{1}{5} ln x^{5} - 1 + C

Solution steps

\int \frac{x ^{14}}{x ^{5} - 1} d x

Long division

\frac{x ^{14}}{x ^{5} - 1} : x^{9} + x^{4} + \frac{x ^{4}}{x ^{5} - 1}

= \int x^{9} + x^{4} + \frac{x ^{4}}{x ^{5} - 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^{9} d x + \int x^{4} d x + \int \frac{x ^{4}}{x ^{5} - 1} d x

\int x^{9} d x = \frac{x ^{10}}{10}

\int x^{4} d x = \frac{x ^{5}}{5}

\int \frac{x ^{4}}{x ^{5} - 1} d x = \frac{1}{5} ln x^{5} - 1

= \frac{x ^{10}}{10} + \frac{x ^{5}}{5} + \frac{1}{5} ln x^{5} - 1

Add a constant to the solution

= \frac{x ^{10}}{10} + \frac{x ^{5}}{5} + \frac{1}{5} ln x^{5} - 1 + C

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\int (e^{3 x} - 4)^{2} d x

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What is the integral of (x^{14})/(x^5-1) ?
The integral of (x^{14})/(x^5-1) is (x^{10}}{10}+\frac{x^5)/5+1/5 ln|x^5-1|+C