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

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

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

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

\frac{x ^{2}}{2} - x + 5 ln ∣ x ∣ + C

Solution steps

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

Expand

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

= \int x - 1 + \frac{5}{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{5}{x} d x

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

\int 1 d x = x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{x ^{2}}{2} - x + 5 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - x + 5 ln ∣ x ∣ + C

n = 1 \sum \infty \frac{n}{n + 5}

\frac{d}{d x} (- \frac{4}{3} sin (x) - \frac{5}{3} cos (x))

x \to + 3 lim (x^{2} + 4)

\frac{\partial}{\partial x} (\frac{a x + b y}{( c x + d y )})

\frac{d}{d x} (tan (arctan (x)))

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