What is the integral of (x^4+1)/(x^3-x) ?

The integral of (x^4+1)/(x^3-x) is (x^2)/2-ln|x|+ln|x+1|+ln|x-1|+C

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integral of (x^4+1)/(x^3-x)

\int \frac{x ^{4} + 1}{x ^{3} - x} d x

\frac{x ^{2}}{2} - ln ∣ x ∣ + ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{x ^{4} + 1}{x ^{3} - x} d x

Long division

\frac{x ^{4} + 1}{x ^{3} - x} : x + \frac{x ^{2} + 1}{x ^{3} - x}

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

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

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

= \frac{x ^{2}}{2} - ln ∣ x ∣ + ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - ln ∣ x ∣ + ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + C

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What is the integral of (x^4+1)/(x^3-x) ?
The integral of (x^4+1)/(x^3-x) is (x^2)/2-ln|x|+ln|x+1|+ln|x-1|+C