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

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

y^{^{''}} - \frac{5}{x} y^{^{'}} + \frac{8}{x ^{2}} y = 0

7 t \frac{d y}{d t} - 3 y = t

y = 4 - x^{2}, (3, - 5)

\frac{d}{d x} (\frac{3}{( 4 x ^{2} - 1 ) ^{4}})

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