What is the integral of (x^5+x^2+2)/(x^3-x) ?

The integral of (x^5+x^2+2)/(x^3-x) is (x^3)/3+x-2ln|x|+ln|x+1|+2ln|x-1|+C

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integral of (x^5+x^2+2)/(x^3-x)

\int \frac{x ^{5} + x ^{2} + 2}{x ^{3} - x} d x

\frac{x ^{3}}{3} + x - 2 ln ∣ x ∣ + ln ∣ x + 1 ∣ + 2 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{x ^{5} + x ^{2} + 2}{x ^{3} - x} d x

Long division

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

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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

\int e^{- 3 x^{- 5}} x^{- 6} d x

\int sin^{3} (x) (cos^{2} (x)) d x

e^{- x}, - 0.75

f^{^{''}} (x) = x

\frac{s - 2}{s ^{2} - 4 s + 5}

What is the integral of (x^5+x^2+2)/(x^3-x) ?
The integral of (x^5+x^2+2)/(x^3-x) is (x^3)/3+x-2ln|x|+ln|x+1|+2ln|x-1|+C