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

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

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

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

x^{2} - x + 3 ln ∣ x ∣ + C

Solution steps

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

Expand

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

= \int 2 x - 1 + \frac{3}{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 2 x d x - \int 1 d x + \int \frac{3}{x} d x

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

\int 1 d x = x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

= x^{2} - x + 3 ln ∣ x ∣

Add a constant to the solution

= x^{2} - x + 3 ln ∣ x ∣ + C

h \to 0 lim (\frac{sin ( h )}{h})

y^{^{''}} - y = t

\int \frac{- 1}{sin ( x )} d x

\int_{0}^{4} x^{\frac{1}{2}} d x

y^{^{'}} x + 2 y = - 2 e^{x}

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