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

The integral of (x+2)/(2x^2-x) is -2ln|x|+5/2 ln|2x-1|+C

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

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

- 2 ln ∣ x ∣ + \frac{5}{2} ln ∣ 2 x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \int - \frac{2}{x} + \frac{5}{2 x - 1} 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 \frac{2}{x} d x + \int \frac{5}{2 x - 1} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{5}{2 x - 1} d x = \frac{5}{2} ln ∣ 2 x - 1 ∣

= - 2 ln ∣ x ∣ + \frac{5}{2} ln ∣ 2 x - 1 ∣

Add a constant to the solution

= - 2 ln ∣ x ∣ + \frac{5}{2} ln ∣ 2 x - 1 ∣ + C

\int e^{(\frac{x ^{2}}{2})} d x

x \to 7 + lim (3)

\frac{\partial}{\partial x} (e^{x^{2} + y^{2} + 1})

x \to \infty lim (\frac{1}{( 2 x + 1 ) ^{\frac{1}{x}}})

x \to \infty lim (x e)

What is the integral of (x+2)/(2x^2-x) ?
The integral of (x+2)/(2x^2-x) is -2ln|x|+5/2 ln|2x-1|+C