What is the integral of (x^3-x-1)/(x^2) ?

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

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integral of (x^3-x-1)/(x^2)

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

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

Solution steps

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

Expand

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Simplify

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

Add a constant to the solution

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

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