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

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

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

\int (\frac{x ^{2} - x + 1}{x ^{2}}) d x

x - ln ∣ x ∣ - \frac{1}{x} + C

Solution steps

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

Expand

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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

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

= x - ln ∣ x ∣ - \frac{1}{x}

Add a constant to the solution

= x - ln ∣ x ∣ - \frac{1}{x} + C

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