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

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

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

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

\frac{x ^{2}}{2} - x + ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{x ^{2}}{2} - x + ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - x + ln ∣ x ∣ + C

(x - 9) y^{^{'}} + y = x^{2} - 10

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

x \to 7 lim (2)

f (x) = \frac{5}{x}, at x = 3

\int 36 x (9 x^{2} - 7)^{3} d x

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